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## Article 39Explanation of Tables.

In Table I the numerical values of the hyperbolic functions $sinhu,coshu,tanhu$ are tabulated for values of $u$ increasing from 0 to 4 at intervals of .02. When $u$ exceeds 4, Table IV may be used.

Table II gives hyperbolic functions of complex arguments, in which

and the values of $a,b,c,d$ are tabulated for values of $x$ and of $y$ ranging separately from 0 to 1.5 at intervals of .1. When interpolation is necessary it may be performed in three stages. For example, to ﬁnd : First ﬁnd , by keeping $y$ at 1.3 and interpolating between the entries under $x=.8$ and $x=.9$; next ﬁnd , by keeping $y$ at 1.4 and interpolating between the entries under $x=.8$ and $x=.9$, as before; then by interpolation between and ﬁnd , in which $x$ is kept at .82. The table is available for all values of $y$, however great, by means of the formulas on page §:

It does not apply when $x$ is greater than 1.5, but this case seldom occurs in practice. This table can also be used as a complex table of circular functions, for

and, moreover, the exponential function is given by

in which the signs of $c$ and $d$ are to be taken the same as the sign of $x$, and the sign of $i$ on the right is to be the product of the signs of $x$ and of $i$ on the left.

Table III gives the values of $v=gdu$, and of the gudermanian angle $\theta =\frac{18{0}^{\circ }v}{\pi }$, as $u$ changes from 0 to 1 at intervals of .02, from 1 to 2 at intervals of .05, and from 2 to 4 at intervals of .1.

In Table IV are given, the values of $gdu$, $logsinhu$, $logcoshu$, as $u$ increases from 4 to 6 at intervals of .1, from 6 to 7 at intervals of .2, and from 7 to 9 at intervals of .5.

In the rare cases in which more extensive tables are necessary, reference may be made to the tables32 of Gudermann, Glaisher, and Geipel and Kilgour. In the ﬁrst the Gudermanian angle (written $k$) is taken as the independent variable, and increases from 0 to 100 grades at intervals of .01, the corresponding value of $u$ (written $Lk$) being tabulated. In the usual case, in which the table is entered with the value of $u$, it gives by interpolation the value of the gudermanian angle, whose circular functions would then give the hyperbolic functions of $u$. When $u$ is large, this angle is so nearly right that interpolation is not reliable. To remedy this inconvenience Gudermann’s second table gives directly $logsinhu$, $logcoshu$, $logtanhu$, to nine ﬁgures, for values of $u$ varying by .001 from 2 to 5, and by .01 from 5 to 12.

Glaisher has tabulated the values of ${e}^{x}$ and ${e}^{-x}$, to nine signiﬁcant ﬁgures, as $x$ varies by .001 from 0 to .1, by .01 from 0 to 2, by .1 from 0 to 10, and by 1 from 0 to 500. From these the values of $coshx$, $sinhx$ are easily obtained.

Geipel and Kilgour’s handbook gives the values of $coshx$, $sinhx$, to seven ﬁgures, as $x$ varies by .01 from 0 to 4.

There are also extensive tables by Forti, Gronau, Vassal, Callet, and Hoüel; and there are four-place tables in Byerly’s Fourier Series, and in Wheeler’s Trigonometry.

In the following tables a dash over a ﬁnal digit indicates that the number has been increased.

TABLE I.—HYPERBOLIC FUNCTIONS.

 $u$ $sinhu.$ $coshu.$ $tanhu.$ $u$ $sinhu.$ $coshu.$ $tanhu.$ $.00$ $.0000$ $1.0000$ $.0000$ $1.00$ $1.1752$ $1.543\stackrel{̄}{1}$ $.7616$ $02$ $0200$ $1.0002$ $0200$ $1.02$ $1.206\stackrel{̄}{3}$ $1.566\stackrel{̄}{9}$ $769\stackrel{̄}{9}$ $04$ $0400$ $1.0008$ $040\stackrel{̄}{0}$ $1.04$ $1.237\stackrel{̄}{9}$ $1.5913$ $777\stackrel{̄}{9}$ $06$ $0600$ $1.0018$ $0599$ $1.06$ $1.270\stackrel{̄}{0}$ $1.6164$ $785\stackrel{̄}{7}$ $08$ $080\stackrel{̄}{1}$ $1.0032$ $0798$ $1.08$ $1.3025$ $1.6421$ $793\stackrel{̄}{2}$ $.10$ $.100\stackrel{̄}{2}$ $1.0050$ $.099\stackrel{̄}{7}$ $1.10$ $1.3356$ $1.6685$ $.8005$ $12$ $120\stackrel{̄}{3}$ $1.0072$ $1194$ $1.12$ $1.369\stackrel{̄}{3}$ $1.695\stackrel{̄}{6}$ $807\stackrel{̄}{6}$ $14$ $140\stackrel{̄}{5}$ $1.0098$ $139\stackrel{̄}{1}$ $1.14$ $1.403\stackrel{̄}{5}$ $1.7233$ $8144$ $16$ $160\stackrel{̄}{7}$ $1.0128$ $1586$ $1.16$ $1.4382$ $1.7517$ $8210$ $18$ $181\stackrel{̄}{0}$ $1.0162$ $178\stackrel{̄}{1}$ $1.18$ $1.4735$ $1.7808$ $827\stackrel{̄}{5}$ $.20$ $.2013$ $1.020\stackrel{̄}{1}$ $.197\stackrel{̄}{4}$ $1.20$ $1.509\stackrel{̄}{5}$ $1.810\stackrel{̄}{7}$ $.833\stackrel{̄}{7}$ $22$ $221\stackrel{̄}{8}$ $1.024\stackrel{̄}{3}$ $2165$ $1.22$ $1.546\stackrel{̄}{0}$ $1.8412$ $839\stackrel{̄}{7}$ $24$ $2423$ $1.0289$ $235\stackrel{̄}{5}$ $1.24$ $1.5831$ $1.872\stackrel{̄}{5}$ $845\stackrel{̄}{5}$ $26$ $2629$ $1.034\stackrel{̄}{0}$ $254\stackrel{̄}{3}$ $1.26$ $1.620\stackrel{̄}{9}$ $1.9045$ $851\stackrel{̄}{1}$ $28$ $283\stackrel{̄}{7}$ $1.0395$ $2729$ $1.28$ $1.6593$ $1.9373$ $856\stackrel{̄}{5}$ $.30$ $.3045$ $1.0453$ $.2913$ $1.30$ $1.6984$ $1.9709$ $.8617$ $32$ $325\stackrel{̄}{5}$ $1.0516$ $3095$ $1.32$ $1.7381$ $2.005\stackrel{̄}{3}$ $8668$ $34$ $3466$ $1.058\stackrel{̄}{4}$ $327\stackrel{̄}{5}$ $1.34$ $1.778\stackrel{̄}{6}$ $2.0404$ $871\stackrel{̄}{7}$ $36$ $3678$ $1.0655$ $3452$ $1.36$ $1.819\stackrel{̄}{8}$ $2.0764$ $876\stackrel{̄}{4}$ $38$ $3892$ $1.0731$ $3627$ $1.38$ $1.861\stackrel{̄}{7}$ $2.1132$ $881\stackrel{̄}{0}$ $.40$ $.410\stackrel{̄}{8}$ $1.081\stackrel{̄}{1}$ $.3799$ $1.40$ $1.9043$ $2.150\stackrel{̄}{9}$ $.8854$ $42$ $432\stackrel{̄}{5}$ $1.0895$ $3969$ $1.42$ $1.9477$ $2.1894$ $889\stackrel{̄}{6}$ $44$ $4543$ $1.098\stackrel{̄}{4}$ $4136$ $1.44$ $1.991\stackrel{̄}{9}$ $2.2288$ $893\stackrel{̄}{7}$ $46$ $476\stackrel{̄}{4}$ $1.107\stackrel{̄}{7}$ $430\stackrel{̄}{1}$ $1.46$ $2.036\stackrel{̄}{9}$ $2.269\stackrel{̄}{1}$ $897\stackrel{̄}{7}$ $48$ $4986$ $1.1174$ $4462$ $1.48$ $2.082\stackrel{̄}{7}$ $2.310\stackrel{̄}{3}$ $901\stackrel{̄}{5}$ $.50$ $.521\stackrel{̄}{1}$ $1.1276$ $.4621$ $1.50$ $2.129\stackrel{̄}{3}$ $2.3524$ $.9051$ $52$ $543\stackrel{̄}{8}$ $1.138\stackrel{̄}{3}$ $4777$ $1.52$ $2.176\stackrel{̄}{8}$ $2.395\stackrel{̄}{5}$ $908\stackrel{̄}{7}$ $54$ $5666$ $1.149\stackrel{̄}{4}$ $493\stackrel{̄}{0}$ $1.54$ $2.2251$ $2.439\stackrel{̄}{5}$ $9121$ $56$ $5897$ $1.1609$ $508\stackrel{̄}{0}$ $1.56$ $2.2743$ $2.484\stackrel{̄}{5}$ $9154$ $58$ $613\stackrel{̄}{1}$ $1.173\stackrel{̄}{0}$ $522\stackrel{̄}{7}$ $1.58$ $2.324\stackrel{̄}{5}$ $2.530\stackrel{̄}{5}$ $9186$ $.60$ $.636\stackrel{̄}{7}$ $1.185\stackrel{̄}{5}$ $.5370$ $1.60$ $2.375\stackrel{̄}{6}$ $2.577\stackrel{̄}{5}$ $.921\stackrel{̄}{7}$ $62$ $660\stackrel{̄}{5}$ $1.1984$ $5511$ $1.62$ $2.427\stackrel{̄}{6}$ $2.625\stackrel{̄}{5}$ $9246$ $64$ $684\stackrel{̄}{6}$ $1.211\stackrel{̄}{9}$ $564\stackrel{̄}{9}$ $1.64$ $2.480\stackrel{̄}{6}$ $2.674\stackrel{̄}{6}$ $927\stackrel{̄}{5}$ $66$ $709\stackrel{̄}{0}$ $1.2258$ $578\stackrel{̄}{4}$ $1.66$ $2.534\stackrel{̄}{6}$ $2.7247$ $9302$ $68$ $7336$ $1.2402$ $5915$ $1.68$ $2.5896$ $2.776\stackrel{̄}{0}$ $932\stackrel{̄}{9}$ $.70$ $.758\stackrel{̄}{6}$ $1.255\stackrel{̄}{2}$ $.604\stackrel{̄}{4}$ $1.70$ $2.6456$ $2.8283$ $.9354$ $72$ $7838$ $1.270\stackrel{̄}{6}$ $6169$ $1.72$ $2.7027$ $2.881\stackrel{̄}{8}$ $937\stackrel{̄}{9}$ $74$ $8094$ $1.2865$ $6291$ $1.74$ $2.7609$ $2.9364$ $9402$ $76$ $8353$ $1.303\stackrel{̄}{0}$ $641\stackrel{̄}{1}$ $1.76$ $2.820\stackrel{̄}{2}$ $2.9922$ $9425$ $78$ $8615$ $1.3199$ $6527$ $1.78$ $2.8806$ $3.0492$ $944\stackrel{̄}{7}$ $.80$ $.8881$ $1.3374$ $.6640$ $1.80$ $2.942\stackrel{̄}{2}$ $3.107\stackrel{̄}{5}$ $.9468$ $82$ $9150$ $1.355\stackrel{̄}{5}$ $675\stackrel{̄}{1}$ $1.82$ $3.0049$ $3.1669$ $9488$ $84$ $9423$ $1.3740$ $6858$ $1.84$ $3.068\stackrel{̄}{9}$ $3.227\stackrel{̄}{7}$ $950\stackrel{̄}{8}$ $86$ $970\stackrel{̄}{0}$ $1.393\stackrel{̄}{2}$ $696\stackrel{̄}{3}$ $1.86$ $3.1340$ $3.2897$ $952\stackrel{̄}{7}$ $88$ $998\stackrel{̄}{1}$ $1.4128$ $7064$ $1.88$ $3.200\stackrel{̄}{5}$ $3.3530$ $954\stackrel{̄}{5}$ $.90$ $1.0265$ $1.433\stackrel{̄}{1}$ $.716\stackrel{̄}{3}$ $1.90$ $3.268\stackrel{̄}{2}$ $3.4177$ $.9562$ $92$ $1.0554$ $1.4539$ $725\stackrel{̄}{9}$ $1.92$ $3.337\stackrel{̄}{2}$ $3.483\stackrel{̄}{8}$ $9579$ $94$ $1.084\stackrel{̄}{7}$ $1.4753$ $7352$ $1.94$ $3.4075$ $3.5512$ $9595$ $96$ $1.1144$ $1.497\stackrel{̄}{3}$ $744\stackrel{̄}{3}$ $1.96$ $3.4792$ $3.620\stackrel{̄}{1}$ $961\stackrel{̄}{1}$ $98$ $1.144\stackrel{̄}{6}$ $1.519\stackrel{̄}{9}$ $753\stackrel{̄}{1}$ $1.98$ $3.5523$ $3.6904$ $962\stackrel{̄}{6}$ $2.00$ $3.626\stackrel{̄}{9}$ $3.762\stackrel{̄}{2}$ $.9640$ $3.00$ $10.017\stackrel{̄}{9}$ $10.067\stackrel{̄}{7}$ $.99505$ $2.02$ $3.7028$ $3.835\stackrel{̄}{5}$ $9654$ $3.02$ $10.2212$ $10.2700$ $99524$ $2.04$ $3.780\stackrel{̄}{3}$ $3.9103$ $9667$ $3.04$ $10.4287$ $10.4765$ $99543$ $2.06$ $3.859\stackrel{̄}{3}$ $3.9867$ $9680$ $3.06$ $10.6403$ $10.6872$ $99561$ $2.08$ $3.9398$ $4.0647$ $969\stackrel{̄}{3}$ $3.08$ $10.8562$ $10.902\stackrel{̄}{2}$ $99578$ $2.10$ $4.021\stackrel{̄}{9}$ $4.1443$ $.970\stackrel{̄}{5}$ $3.10$ $11.076\stackrel{̄}{5}$ $11.1215$ $.99594$ $2.12$ $4.1055$ $4.225\stackrel{̄}{6}$ $971\stackrel{̄}{6}$ $3.12$ $11.3011$ $11.345\stackrel{̄}{3}$ $99610$ $2.14$ $4.190\stackrel{̄}{9}$ $4.3085$ $972\stackrel{̄}{7}$ $3.14$ $11.530\stackrel{̄}{3}$ $11.573\stackrel{̄}{6}$ $99626$ $2.16$ $4.2779$ $4.3932$ $9737$ $3.16$ $11.764\stackrel{̄}{1}$ $11.8065$ $99640$ $2.18$ $4.3666$ $4.479\stackrel{̄}{7}$ $974\stackrel{̄}{8}$ $3.18$ $12.002\stackrel{̄}{6}$ $12.044\stackrel{̄}{2}$ $99654$ $2.20$ $4.4571$ $4.5679$ $.9757$ $3.20$ $12.245\stackrel{̄}{9}$ $12.2866$ $.99668$ $2.22$ $4.549\stackrel{̄}{4}$ $4.658\stackrel{̄}{0}$ $976\stackrel{̄}{7}$ $3.22$ $12.494\stackrel{̄}{1}$ $12.5340$ $99681$ $2.24$ $4.6434$ $4.749\stackrel{̄}{9}$ $977\stackrel{̄}{6}$ $3.24$ $12.747\stackrel{̄}{3}$ $12.7864$ $99693$ $2.26$ $4.739\stackrel{̄}{4}$ $4.8437$ $978\stackrel{̄}{5}$ $3.26$ $13.005\stackrel{̄}{6}$ $13.044\stackrel{̄}{0}$ $99705$ $2.28$ $4.837\stackrel{̄}{2}$ $4.939\stackrel{̄}{5}$ $979\stackrel{̄}{3}$ $3.28$ $13.269\stackrel{̄}{1}$ $13.3067$ $99717$ $2.30$ $4.937\stackrel{̄}{0}$ $5.0372$ $.980\stackrel{̄}{1}$ $3.30$ $13.537\stackrel{̄}{9}$ $13.574\stackrel{̄}{8}$ $.99728$ $2.32$ $5.0387$ $5.137\stackrel{̄}{0}$ $980\stackrel{̄}{9}$ $3.32$ $13.812\stackrel{̄}{1}$ $13.848\stackrel{̄}{3}$ $99738$ $2.34$ $5.1425$ $5.238\stackrel{̄}{8}$ $9816$ $3.34$ $14.0918$ $14.127\stackrel{̄}{3}$ $99749$ $2.36$ $5.248\stackrel{̄}{3}$ $5.342\stackrel{̄}{7}$ $9823$ $3.36$ $14.3772$ $14.412\stackrel{̄}{0}$ $99758$ $2.38$ $5.356\stackrel{̄}{2}$ $5.4487$ $9830$ $3.38$ $14.668\stackrel{̄}{4}$ $14.7024$ $99768$ $2.40$ $5.4662$ $5.5569$ $.983\stackrel{̄}{7}$ $3.40$ $14.965\stackrel{̄}{4}$ $14.9987$ $.99777$ $2.42$ $5.5785$ $5.667\stackrel{̄}{4}$ $9843$ $3.42$ $15.268\stackrel{̄}{4}$ $15.301\stackrel{̄}{1}$ $99786$ $2.44$ $5.6929$ $5.7801$ $9849$ $3.44$ $15.5774$ $15.6095$ $99794$ $2.46$ $5.809\stackrel{̄}{7}$ $5.8951$ $9855$ $3.46$ $15.892\stackrel{̄}{8}$ $15.9242$ $99802$ $2.48$ $5.928\stackrel{̄}{8}$ $6.0125$ $986\stackrel{̄}{1}$ $3.48$ $16.2144$ $16.245\stackrel{̄}{3}$ $99810$ $2.50$ $6.0502$ $6.132\stackrel{̄}{3}$ $.9866$ $3.50$ $16.5426$ $16.5728$ $.99817$ $2.52$ $6.174\stackrel{̄}{1}$ $6.2545$ $9871$ $3.52$ $16.8774$ $16.9070$ $99824$ $2.54$ $6.3004$ $6.379\stackrel{̄}{3}$ $9876$ $3.54$ $17.219\stackrel{̄}{0}$ $17.248\stackrel{̄}{0}$ $99831$ $2.56$ $6.429\stackrel{̄}{3}$ $6.506\stackrel{̄}{6}$ $9881$ $3.56$ $17.567\stackrel{̄}{4}$ $17.5958$ $99831$ $2.58$ $6.560\stackrel{̄}{7}$ $6.6364$ $988\stackrel{̄}{6}$ $3.58$ $17.9228$ $17.9507$ $99844$ $2.60$ $6.6947$ $6.7690$ $.9890$ $3.60$ $18.2854$ $18.312\stackrel{̄}{8}$ $.99850$ $2.62$ $6.831\stackrel{̄}{5}$ $6.904\stackrel{̄}{3}$ $989\stackrel{̄}{5}$ $3.62$ $18.655\stackrel{̄}{4}$ $18.682\stackrel{̄}{2}$ $99856$ $2.64$ $6.9709$ $7.042\stackrel{̄}{3}$ $989\stackrel{̄}{9}$ $3.64$ $19.032\stackrel{̄}{8}$ $19.0590$ $99862$ $2.66$ $7.113\stackrel{̄}{2}$ $7.183\stackrel{̄}{2}$ $990\stackrel{̄}{3}$ $3.66$ $19.4178$ $19.4435$ $99867$ $2.68$ $7.258\stackrel{̄}{3}$ $7.3268$ $9906$ $3.68$ $19.810\stackrel{̄}{6}$ $19.8358$ $99872$ $2.70$ $7.406\stackrel{̄}{3}$ $7.473\stackrel{̄}{5}$ $.9910$ $3.70$ $20.211\stackrel{̄}{3}$ $20.2360$ $.99877$ $2.72$ $7.5572$ $7.623\stackrel{̄}{1}$ $991\stackrel{̄}{4}$ $3.72$ $20.620\stackrel{̄}{1}$ $20.6443$ $99882$ $2.74$ $7.7112$ $7.775\stackrel{̄}{8}$ $991\stackrel{̄}{7}$ $3.74$ $21.0371$ $21.060\stackrel{̄}{9}$ $99887$ $2.76$ $7.868\stackrel{̄}{3}$ $7.931\stackrel{̄}{6}$ $9920$ $3.76$ $21.462\stackrel{̄}{6}$ $21.485\stackrel{̄}{9}$ $99891$ $2.78$ $8.028\stackrel{̄}{5}$ $8.0905$ $9923$ $3.78$ $21.8966$ $21.9194$ $9989\stackrel{̄}{6}$ $2.80$ $8.1919$ $8.2527$ $.9926$ $3.80$ $22.3394$ $22.361\stackrel{̄}{8}$ $.9990\stackrel{̄}{0}$ $2.82$ $8.3586$ $8.4182$ $9929$ $3.82$ $22.7911$ $22.813\stackrel{̄}{1}$ $9990\stackrel{̄}{4}$ $2.84$ $8.528\stackrel{̄}{7}$ $8.587\stackrel{̄}{1}$ $993\stackrel{̄}{2}$ $3.84$ $23.252\stackrel{̄}{0}$ $23.273\stackrel{̄}{5}$ $99907$ $2.86$ $8.7021$ $8.759\stackrel{̄}{4}$ $993\stackrel{̄}{5}$ $3.86$ $23.7221$ $23.7432$ $99911$ $2.88$ $8.879\stackrel{̄}{1}$ $8.9352$ $9937$ $3.88$ $24.2018$ $24.2224$ $9991\stackrel{̄}{5}$ $2.90$ $9.059\stackrel{̄}{6}$ $9.114\stackrel{̄}{6}$ $.994\stackrel{̄}{0}$ $3.90$ $24.6911$ $24.7113$ $.99918$ $2.92$ $9.243\stackrel{̄}{7}$ $9.2976$ $994\stackrel{̄}{2}$ $3.92$ $25.1903$ $25.2101$ $99921$ $2.94$ $9.431\stackrel{̄}{5}$ $9.484\stackrel{̄}{4}$ $9944$ $3.94$ $25.699\stackrel{̄}{6}$ $25.7190$ $99924$ $2.96$ $9.623\stackrel{̄}{1}$ $9.674\stackrel{̄}{9}$ $994\stackrel{̄}{7}$ $3.96$ $26.2191$ $26.238\stackrel{̄}{2}$ $99927$ $2.98$ $9.8185$ $9.8693$ $994\stackrel{̄}{9}$ $3.98$ $26.749\stackrel{̄}{2}$ $26.767\stackrel{̄}{9}$ $99930$

TABLE II.—VALUES OF AND .

 $x=0$ $x=.1$ $y$ $a$ $b$ $c$ $d$ $a$ $b$ $c$ $d$ 0 $1.0000$ $0000$ $0000$ $.0000$ $1.0050$ $.00000$ $.1001\stackrel{̄}{7}$ $.0000$ .1 $0.9950$ ” ” $0998$ $1.000\stackrel{̄}{0}$ $01000$ $09967$ $1003$ .2 $0.980\stackrel{̄}{1}$ ” ” $198\stackrel{̄}{7}$ $0.9850$ $0199\stackrel{̄}{0}$ $09817$ $199\stackrel{̄}{7}$ .3 $0.9553$ ” ” $2955$ $0.9601$ $02960$ $0957\stackrel{̄}{0}$ $297\stackrel{̄}{0}$ .4 $.921\stackrel{̄}{1}$ ” ” $.3894$ $.925\stackrel{̄}{7}$ $.03901$ $.09226$ $.3914$ .5 $8776$ ” ” $4794$ $882\stackrel{̄}{0}$ $04802$ $0879\stackrel{̄}{1}$ $4818$ .6 $8253$ ” ” $5646$ $829\stackrel{̄}{5}$ $05656$ $08267$ $567\stackrel{̄}{5}$ .7 $7648$ ” ” $6442$ $768\stackrel{̄}{7}$ $06453$ $07661$ $6474$ .8 $.6967$ ” ” $.717\stackrel{̄}{4}$ $.700\stackrel{̄}{2}$ $.0718\stackrel{̄}{6}$ $.0697\stackrel{̄}{9}$ $.7800$ .9 $6216$ ” ” $7833$ $624\stackrel{̄}{7}$ $0784\stackrel{̄}{7}$ $0622\stackrel{̄}{7}$ $7872$ 1.0 $5403$ ” ” $841\stackrel{̄}{5}$ $5430$ $08429$ $05412$ $845\stackrel{̄}{7}$ 1.1 $4536$ ” ” $8912$ $455\stackrel{̄}{9}$ $08927$ $04544$ $895\stackrel{̄}{7}$ 1.2 $.362\stackrel{̄}{4}$ ” ” $.9320$ $.364\stackrel{̄}{2}$ $.09336$ $.0363\stackrel{̄}{0}$ $0.936\stackrel{̄}{7}$ 1.3 $2675$ ” ” $963\stackrel{̄}{6}$ $268\stackrel{̄}{8}$ $0965\stackrel{̄}{2}$ $0268\stackrel{̄}{0}$ $0.968\stackrel{̄}{4}$ 1.4 $170\stackrel{̄}{0}$ ” ” $9854$ $1708$ $09871$ $0170\stackrel{̄}{3}$ $0.990\stackrel{̄}{4}$ 1.5 $0707$ ” ” $997\stackrel{̄}{5}$ $0711$ $0999\stackrel{̄}{2}$ $0070\stackrel{̄}{9}$ $1.002\stackrel{̄}{5}$ $\frac{1}{2}\pi$ $0000$ ” ” $1.0000$ $0000$ $1001\stackrel{̄}{7}$ $00000$ $1.0050$

 $x=.2$ $x=.3$ $y$ $a$ $b$ $c$ $d$ $a$ $b$ $c$ $d$ 0 $1.020\stackrel{̄}{1}$ $.0000$ $.2013$ $.0000$ $1.0453$ $.0000$ $.3045$ $.0000$ .1 $1.015\stackrel{̄}{0}$ $0201$ $2003$ $1018$ $1.040\stackrel{̄}{1}$ $0304$ $303\stackrel{̄}{0}$ $1044$ .2 $0.9997$ $0400$ $1973$ $202\stackrel{̄}{7}$ $1.024\stackrel{̄}{5}$ $0605$ $298\stackrel{̄}{5}$ $207\stackrel{̄}{7}$ .3 $0.9745$ $0595$ $1923$ $3014$ $9987$ $090\stackrel{̄}{0}$ $2909$ $3089$ .4 $.9395$ $.0784$ $.1854$ $.3972$ $.9628$ $.1186$ $.280\stackrel{̄}{5}$ $.407\stackrel{̄}{1}$ .5 $895\stackrel{̄}{2}$ $0965$ $176\stackrel{̄}{7}$ $4890$ $917\stackrel{̄}{4}$ $146\stackrel{̄}{0}$ $267\stackrel{̄}{2}$ $501\stackrel{̄}{2}$ .6 $8419$ $113\stackrel{̄}{7}$ $166\stackrel{̄}{2}$ $576\stackrel{̄}{0}$ $8687$ $1719$ $2513$ $590\stackrel{̄}{3}$ .7 $780\stackrel{̄}{2}$ $1297$ $154\stackrel{̄}{0}$ $6571$ $7995$ $196\stackrel{̄}{2}$ $2329$ $6734$ .8 $.710\stackrel{̄}{7}$ $.1444$ $.140\stackrel{̄}{3}$ $.731\stackrel{̄}{8}$ $.728\stackrel{̄}{3}$ $.2184$ $.212\stackrel{̄}{2}$ $.7498$ .9 $634\stackrel{̄}{1}$ $1577$ $125\stackrel{̄}{2}$ $7990$ $649\stackrel{̄}{8}$ $2385$ $189\stackrel{̄}{3}$ $8188$ 1.0 $5511$ $1694$ $108\stackrel{̄}{8}$ $858\stackrel{̄}{4}$ $5648$ $2562$ $1645$ $8796$ 1.1 $4627$ $179\stackrel{̄}{5}$ $0913$ $909\stackrel{̄}{1}$ $474\stackrel{̄}{2}$ $2714$ $1381$ $9316$ 1.2 $.3696$ $.187\stackrel{̄}{7}$ $.073\stackrel{̄}{0}$ $0.9507$ $.378\stackrel{̄}{8}$ $.2838$ $.1103$ $0.974\stackrel{̄}{3}$ 1.3 $272\stackrel{̄}{9}$ $1940$ $053\stackrel{̄}{9}$ $0.982\stackrel{̄}{9}$ $2796$ $2934$ $081\stackrel{̄}{5}$ $1.0072$ 1.4 $173\stackrel{̄}{4}$ $1984$ $0342$ $1.0052$ $177\stackrel{̄}{7}$ $3001$ $051\stackrel{̄}{8}$ $1.0301$ 1.5 $072\stackrel{̄}{2}$ $2008$ $0142$ $1.0175$ $0739$ $303\stackrel{̄}{8}$ $0215$ $1.042\stackrel{̄}{7}$ $\frac{1}{2}\pi$ $0000$ $2013$ $0000$ $1.020\stackrel{̄}{1}$ $0000$ $3045$ $0000$ $1.0453$

TABLE II.—VALUES OF AND . (continued)

 $x=.4$ $x=.5$ $y$ $a$ $b$ $c$ $d$ $a$ $b$ $c$ $d$ 0 $1.081\stackrel{̄}{1}$ $.0000$ $.410\stackrel{̄}{8}$ $.0000$ $1.1276$ $.0000$ $.521\stackrel{̄}{1}$ $.0000$ .1 $1.0756$ $0410$ $408\stackrel{̄}{7}$ $1079$ $1.122\stackrel{̄}{0}$ $0520$ $518\stackrel{̄}{5}$ $1126$ .2 $1.0595$ $0816$ $402\stackrel{̄}{6}$ $214\stackrel{̄}{8}$ $1.1051$ $1025$ $5107$ $2240$ .3 $1.032\stackrel{̄}{8}$ $121\stackrel{̄}{4}$ $3924$ $319\stackrel{̄}{5}$ $1.077\stackrel{̄}{3}$ $154\stackrel{̄}{0}$ $4978$ $3332$ .4 $.9957$ $.160\stackrel{̄}{0}$ $.3783$ $.421\stackrel{̄}{0}$ $1.0386$ $.2029$ $.480\stackrel{̄}{0}$ $.4391$ .5 $9487$ $1969$ $360\stackrel{̄}{5}$ $518\stackrel{̄}{3}$ $0.989\stackrel{̄}{6}$ $2498$ $4573$ $5406$ .6 $8922$ $2319$ $3390$ $6104$ $0.9306$ $2942$ $430\stackrel{̄}{1}$ $6367$ .7 $8268$ $2646$ $314\stackrel{̄}{2}$ $6964$ $0.8624$ $335\stackrel{̄}{7}$ $398\stackrel{̄}{6}$ $7264$ .8 $.753\stackrel{̄}{2}$ $.2947$ $.286\stackrel{̄}{2}$ $.7755$ $.7856$ $.3738$ $.363\stackrel{̄}{1}$ $0.8089$ .9 $672\stackrel{̄}{0}$ $3218$ $2553$ $8468$ $7009$ $408\stackrel{̄}{2}$ $3239$ $0.8833$ 1.0 $5841$ $3456$ $2219$ $909\stackrel{̄}{7}$ $609\stackrel{̄}{3}$ $438\stackrel{̄}{5}$ $2815$ $0.948\stackrel{̄}{9}$ 1.1 $4904$ $366\stackrel{̄}{1}$ $1863$ $963\stackrel{̄}{5}$ $511\stackrel{̄}{5}$ $4644$ $236\stackrel{̄}{4}$ $1.005\stackrel{̄}{0}$ 1.2 $.3917$ $.328\stackrel{̄}{9}$ $.1488$ $1.0076$ $.4056$ $.485\stackrel{̄}{7}$ $.1888$ $1.051\stackrel{̄}{0}$ 1.3 $289\stackrel{̄}{2}$ $395\stackrel{̄}{8}$ $109\stackrel{̄}{9}$ $1.041\stackrel{̄}{7}$ $3016$ $5021$ $139\stackrel{̄}{4}$ $1.0865$ 1.4 $183\stackrel{̄}{8}$ $404\stackrel{̄}{8}$ $0698$ $1.0653$ $191\stackrel{̄}{7}$ $5135$ $088\stackrel{̄}{6}$ $1.1163$ 1.5 $076\stackrel{̄}{5}$ $4097$ $029\stackrel{̄}{1}$ $1.078\stackrel{̄}{4}$ $079\stackrel{̄}{8}$ $519\stackrel{̄}{8}$ $036\stackrel{̄}{9}$ $1.124\stackrel{̄}{8}$ $\frac{1}{2}\pi$ $0000$ $410\stackrel{̄}{8}$ $0000$ $1.081\stackrel{̄}{1}$ $0000$ $521\stackrel{̄}{1}$ $0000$ $1.1276$
 $x=.6$ $x=.7$ $y$ $a$ $b$ $c$ $d$ $a$ $b$ $c$ $d$ 0 $1.185\stackrel{̄}{5}$ $.0000$ $.636\stackrel{̄}{7}$ $.0000$ $1.2552$ $.0000$ $.758\stackrel{̄}{6}$ $.0000$ .1 $1.1795$ $063\stackrel{̄}{6}$ $633\stackrel{̄}{5}$ $1183$ $1.248\stackrel{̄}{9}$ $0757$ $754\stackrel{̄}{8}$ $1253$ .2 $1.161\stackrel{̄}{8}$ $126\stackrel{̄}{5}$ $624\stackrel{̄}{0}$ $2355$ $1.2301$ $1542$ $743\stackrel{̄}{5}$ $249\stackrel{̄}{4}$ .3 $1.132\stackrel{̄}{5}$ $1881$ $6082$ $3503$ $1.1991$ $224\stackrel{̄}{2}$ $7247$ $3709$ .4 $1.0918$ $.2479$ $.5864$ $.461\stackrel{̄}{7}$ $1.156\stackrel{̄}{1}$ $.2954$ $.6987$ $.488\stackrel{̄}{8}$ .5 $1.0403$ $3052$ $5587$ $5684$ $1.1015$ $363\stackrel{̄}{7}$ $6657$ $601\stackrel{̄}{8}$ .6 $0.9784$ $395\stackrel{̄}{5}$ $525\stackrel{̄}{5}$ $669\stackrel{̄}{4}$ $1.0359$ $4253$ $626\stackrel{̄}{1}$ $7087$ .7 $0.906\stackrel{̄}{7}$ $4101$ $4869$ $763\stackrel{̄}{7}$ $0.960\stackrel{̄}{0}$ $488\stackrel{̄}{7}$ $580\stackrel{̄}{2}$ $8086$ .8 $.8259$ $.4567$ $.443\stackrel{̄}{6}$ $0.8504$ $.874\stackrel{̄}{5}$ $.544\stackrel{̄}{2}$ $.5285$ $0.9004$ .9 $736\stackrel{̄}{9}$ $4987$ $3957$ $0.9286$ $7802$ $5942$ $4715$ $0.9832$ 1.0 $6405$ $5357$ $344\stackrel{̄}{0}$ $0.9975$ $678\stackrel{̄}{2}$ $6383$ $409\stackrel{̄}{9}$ $1.056\stackrel{̄}{2}$ 1.1 $5377$ $567\stackrel{̄}{4}$ $288\stackrel{̄}{8}$ $1.056\stackrel{̄}{5}$ $5693$ $6760$ $344\stackrel{̄}{1}$ $1.1186$ 1.2 $.429\stackrel{̄}{6}$ $.593\stackrel{̄}{4}$ $.230\stackrel{̄}{7}$ $1.104\stackrel{̄}{9}$ $.4548$ $.7070$ $.274\stackrel{̄}{9}$ $1.169\stackrel{̄}{9}$ 1.3 $3171$ $613\stackrel{̄}{5}$ $1703$ $1.1422$ $335\stackrel{̄}{8}$ $7309$ $2029$ $1.2094$ 1.4 $201\stackrel{̄}{5}$ $627\stackrel{̄}{4}$ $1082$ $1.1682$ $2133$ $7475$ $1289$ $1.2369$ 1.5 $083\stackrel{̄}{9}$ $635\stackrel{̄}{1}$ $0450$ $1.182\stackrel{̄}{5}$ $088\stackrel{̄}{8}$ $756\stackrel{̄}{7}$ $053\stackrel{̄}{7}$ $1.2520$ $\frac{1}{2}\pi$ $0000$ $636\stackrel{̄}{7}$ $0000$ $1.185\stackrel{̄}{5}$ $0000$ $7586$ $0000$ $1.2552$

TABLE II.—VALUES OF AND . (continued)

 $x=.8$ $x=.9$ $y$ $a$ $b$ $c$ $d$ $a$ $b$ $c$ $d$ 0 $1.3374$ $.0000$ $.8881$ $.0000$ $1.433\stackrel{̄}{1}$ $.0000$ $1.0265$ $.0000$ .1 $1.330\stackrel{̄}{8}$ $088\stackrel{̄}{7}$ $883\stackrel{̄}{7}$ $1335$ $1.4259$ $102\stackrel{̄}{5}$ $1.021\stackrel{̄}{4}$ $143\stackrel{̄}{1}$ .2 $1.3108$ $1764$ $8704$ $2657$ $1.4045$ $2039$ $1.006\stackrel{̄}{1}$ $2847$ .3 $1.2776$ $262\stackrel{̄}{5}$ $8484$ $3952$ $1.3691$ $303\stackrel{̄}{4}$ $0.980\stackrel{̄}{7}$ $4235$ .4 $1.231\stackrel{̄}{9}$ $.3458$ $.8180$ $.5208$ $1.320\stackrel{̄}{0}$ $.3997$ $.945\stackrel{̄}{5}$ $.558\stackrel{̄}{1}$ .5 $1.173\stackrel{̄}{7}$ $425\stackrel{̄}{8}$ $779\stackrel{̄}{4}$ $641\stackrel{̄}{2}$ $1.257\stackrel{̄}{7}$ $4921$ $9008$ $687\stackrel{̄}{1}$ .6 $1.1038$ $501\stackrel{̄}{5}$ $733\stackrel{̄}{0}$ $755\stackrel{̄}{2}$ $1.182\stackrel{̄}{8}$ $5796$ $8472$ $809\stackrel{̄}{2}$ .7 $1.0229$ $5721$ $679\stackrel{̄}{3}$ $861\stackrel{̄}{6}$ $1.096\stackrel{̄}{1}$ $661\stackrel{̄}{3}$ $7851$ $9232$ .8 $.931\stackrel{̄}{8}$ $.637\stackrel{̄}{1}$ $.618\stackrel{̄}{8}$ $0.9595$ $.9984$ $.736\stackrel{̄}{4}$ $.715\stackrel{̄}{2}$ $1.0280$ .9 $831\stackrel{̄}{4}$ $695\stackrel{̄}{7}$ $552\stackrel{̄}{1}$ $1.0476$ $8908$ $804\stackrel{̄}{1}$ $638\stackrel{̄}{1}$ $1.1226$ 1.0 $7226$ $7472$ $4798$ $1.1254$ $7743$ $8638$ $5546$ $1.205\stackrel{̄}{9}$ 1.1 $606\stackrel{̄}{7}$ $791\stackrel{̄}{5}$ $4028$ $1.1919$ $6500$ $9148$ $4656$ $1.277\stackrel{̄}{2}$ 1.2 $.4846$ $.827\stackrel{̄}{8}$ $.3218$ $1.2465$ $.519\stackrel{̄}{3}$ $0.956\stackrel{̄}{8}$ $.372\stackrel{̄}{0}$ $1.335\stackrel{̄}{7}$ 1.3 $357\stackrel{̄}{8}$ $8557$ $237\stackrel{̄}{6}$ $1.288\stackrel{̄}{7}$ $383\stackrel{̄}{4}$ $0.9891$ $274\stackrel{̄}{6}$ $1.380\stackrel{̄}{9}$ 1.4 $2273$ $875\stackrel{̄}{2}$ $151\stackrel{̄}{0}$ $1.3180$ $2436$ $1.0124$ $1745$ $1.4122$ 1.5 $0946$ $885\stackrel{̄}{9}$ $0628$ $1.334\stackrel{̄}{1}$ $101\stackrel{̄}{4}$ $1.0239$ $0726$ $1.429\stackrel{̄}{5}$ $\frac{1}{2}\pi$ $0000$ $.8881$ $0000$ $1.3374$ $0000$ $1.0265$ $0000$ $1.433\stackrel{̄}{1}$
 $x=1.0$ $x=1.1$ $y$ $a$ $b$ $c$ $d$ $a$ $b$ $c$ $d$ 0 $1.543\stackrel{̄}{1}$ $.0000$ $1.1752$ $.0000$ $1.6685$ $.0000$ $1.3356$ $.0000$ .1 $1.535\stackrel{̄}{4}$ $1173$ $1.1693$ $154\stackrel{̄}{1}$ $1.660\stackrel{̄}{2}$ $1333$ $1.329\stackrel{̄}{0}$ $1666$ .2 $1.5123$ $2335$ $1.1518$ $306\stackrel{̄}{6}$ $1.635\stackrel{̄}{3}$ $2654$ $1.3090$ $331\stackrel{̄}{5}$ .3 $1.474\stackrel{̄}{2}$ $347\stackrel{̄}{3}$ $1.1227$ $4560$ $1.594\stackrel{̄}{0}$ $3946$ $1.276\stackrel{̄}{0}$ $493\stackrel{̄}{1}$ .4 $1.421\stackrel{̄}{3}$ $457\stackrel{̄}{6}$ $1.0824$ $.6009$ $1.5368$ $5201$ $1.2302$ $0.649\stackrel{̄}{8}$ .5 $1.354\stackrel{̄}{2}$ $5634$ $1.031\stackrel{̄}{4}$ $739\stackrel{̄}{8}$ $1.464\stackrel{̄}{3}$ $6403$ $1.1721$ $0.7999$ .6 $1.273\stackrel{̄}{6}$ $663\stackrel{̄}{6}$ $0.9699$ $871\stackrel{̄}{8}$ $1.377\stackrel{̄}{1}$ $754\stackrel{̄}{2}$ $1.102\stackrel{̄}{4}$ $0.9421$ .7 $1.1802$ $757\stackrel{̄}{1}$ $0.8988$ $994\stackrel{̄}{1}$ $1.276\stackrel{̄}{2}$ $8604$ $1.021\stackrel{̄}{6}$ $1.074\stackrel{̄}{9}$ .8 $1.075\stackrel{̄}{1}$ $0.8430$ $.818\stackrel{̄}{8}$ $1.1069$ $1.162\stackrel{̄}{5}$ $0.9581$ $.930\stackrel{̄}{6}$ $1.1969$ .9 $0.9592$ $0.920\stackrel{̄}{6}$ $7305$ $1.2087$ $1.037\stackrel{̄}{2}$ $1.0462$ $8302$ $1.3070$ 1.0 $0.8337$ $0.9889$ $635\stackrel{̄}{0}$ $1.298\stackrel{̄}{5}$ $0.9015$ $1.1239$ $721\stackrel{̄}{7}$ $1.4040$ 1.1 $0.6999$ $1.0473$ $533\stackrel{̄}{1}$ $1.375\stackrel{̄}{2}$ $0.7568$ $1.1903$ $6058$ $1.487\stackrel{̄}{0}$ 1.2 $.559\stackrel{̄}{2}$ $1.0953$ $.4258$ $1.4382$ $.6046$ $1.244\stackrel{̄}{9}$ $.484\stackrel{̄}{0}$ $1.5551$ 1.3 $5128$ $1.132\stackrel{̄}{4}$ $314\stackrel{̄}{4}$ $1.486\stackrel{̄}{8}$ $4463$ $1.287\stackrel{̄}{0}$ $357\stackrel{̄}{5}$ $1.6077$ 1.4 $262\stackrel{̄}{3}$ $1.158\stackrel{̄}{1}$ $199\stackrel{̄}{8}$ $1.5213$ $2836$ $1.3162$ $2270$ $1.6442$ 1.5 $109\stackrel{̄}{2}$ $1.172\stackrel{̄}{3}$ $0831$ $1.5392$ $1180$ $1.332\stackrel{̄}{3}$ $094\stackrel{̄}{5}$ $1.6643$ $\frac{1}{2}\pi$ $0000$ $1.1752$ $0000$ $1.543\stackrel{̄}{1}$ $.0000$ $1.3356$ $.0000$ $1.6685$

TABLE II.—VALUES OF AND . (continued)

 $x=1.2$ $x=1.3$ $y$ $a$ $b$ $c$ $d$ $a$ $b$ $c$ $d$ 0 $1.810\stackrel{̄}{7}$ $.0000$ $1.509\stackrel{̄}{5}$ $.0000$ $1.9709$ $0000$ $1.698\stackrel{̄}{4}$ $.0000$ .1 $1.8016$ $150\stackrel{̄}{7}$ $1.5019$ $180\stackrel{̄}{8}$ $1.961\stackrel{̄}{1}$ $169\stackrel{̄}{6}$ $1.689\stackrel{̄}{9}$ $196\stackrel{̄}{8}$ .2 $1.774\stackrel{̄}{6}$ $299\stackrel{̄}{9}$ $1.479\stackrel{̄}{4}$ $359\stackrel{̄}{8}$ $1.9316$ $3374$ $1.6645$ $3916$ .3 $1.729\stackrel{̄}{8}$ $446\stackrel{̄}{1}$ $1.4420$ $535\stackrel{̄}{1}$ $1.882\stackrel{̄}{9}$ $5019$ $1.6225$ $5824$ .4 $1.6677$ $.5878$ $1.3903$ $0.7051$ $1.8153$ $.661\stackrel{̄}{4}$ $1.5643$ $0.7675$ .5 $1.5890$ $723\stackrel{̄}{7}$ $1.324\stackrel{̄}{7}$ $0.868\stackrel{̄}{1}$ $1.7296$ $8142$ $1.490\stackrel{̄}{5}$ $0.9449$ .6 $1.4944$ $8523$ $1.2458$ $1.022\stackrel{̄}{4}$ $1.626\stackrel{̄}{7}$ $959\stackrel{̄}{0}$ $1.4017$ $1.1131$ .7 $1.384\stackrel{̄}{9}$ $9724$ $1.154\stackrel{̄}{5}$ $1.166\stackrel{̄}{5}$ $1.5074$ $1.0941$ $1.299\stackrel{̄}{0}$ $1.2697$ .8 $1.261\stackrel{̄}{5}$ $1.0828$ $1.051\stackrel{̄}{7}$ $1.298\stackrel{̄}{9}$ $1.3731$ $1.2183$ $1.183\stackrel{̄}{3}$ $1.413\stackrel{̄}{9}$ .9 $1.1255$ $1.182\stackrel{̄}{4}$ $0.938\stackrel{̄}{3}$ $1.4183$ $1.2251$ $1.330\stackrel{̄}{4}$ $1.0557$ $1.543\stackrel{̄}{9}$ 1.0 $0.9783$ $1.270\stackrel{̄}{2}$ $0.815\stackrel{̄}{6}$ $1.5236$ $1.064\stackrel{̄}{9}$ $1.4291$ $0.9176$ $1.658\stackrel{̄}{5}$ 1.1 $0.8213$ $1.3452$ $0.684\stackrel{̄}{7}$ $1.613\stackrel{̄}{7}$ $0.8940$ $1.5136$ $0.770\stackrel{̄}{4}$ $1.756\stackrel{̄}{5}$ 1.2 $.6561$ $1.406\stackrel{̄}{9}$ $0.547\stackrel{̄}{0}$ $1.6876$ $.714\stackrel{̄}{2}$ $1.583\stackrel{̄}{0}$ $0.6154$ $1.837\stackrel{̄}{0}$ 1.3 $484\stackrel{̄}{4}$ $1.4544$ $0.403\stackrel{̄}{8}$ $1.744\stackrel{̄}{7}$ $5272$ $1.636\stackrel{̄}{5}$ $0.4543$ $1.899\stackrel{̄}{1}$ 1.4 $307\stackrel{̄}{8}$ $1.487\stackrel{̄}{5}$ $0.256\stackrel{̄}{6}$ $1.7843$ $3350$ $1.673\stackrel{̄}{7}$ $0.288\stackrel{̄}{7}$ $1.9422$ 1.5 $128\stackrel{̄}{1}$ $1.505\stackrel{̄}{7}$ $0.106\stackrel{̄}{8}$ $1.8061$ $1394$ $1.6941$ $0.1201$ $1.966\stackrel{̄}{0}$ $\frac{1}{2}\pi$ $0000$ $1.509\stackrel{̄}{5}$ $0000$ $1.810\stackrel{̄}{7}$ $0000$ $1.698\stackrel{̄}{4}$ $0000$ $1.9709$
 $x=1.4$ $x=1.5$ $y$ $a$ $b$ $c$ $d$ $a$ $b$ $c$ $d$ 0 $2.150\stackrel{̄}{9}$ $.0000$ $1.9043$ $.0000$ $2.3524$ $.0000$ $2.129\stackrel{̄}{3}$ $.0000$ .1 $2.1401$ $1901$ $1.8948$ $2147$ $2.3413$ $2126$ $2.118\stackrel{̄}{7}$ $2348$ .2 $2.1080$ $3783$ $1.8663$ $4273$ $2.3055$ $4230$ $2.0868$ $4674$ .3 $2.0548$ $562\stackrel{̄}{8}$ $1.8192$ $6356$ $2.2473$ $6292$ $2.034\stackrel{̄}{2}$ $6951$ .4 $1.9811$ $0.741\stackrel{̄}{6}$ $1.7540$ $0.8376$ $2.1667$ $0.829\stackrel{̄}{2}$ $1.961\stackrel{̄}{2}$ $0.916\stackrel{̄}{1}$ .5 $1.887\stackrel{̄}{6}$ $0.913\stackrel{̄}{0}$ $1.671\stackrel{̄}{2}$ $1.031\stackrel{̄}{2}$ $2.0644$ $1.0208$ $1.8686$ $1.1278$ .6 $1.7752$ $1.075\stackrel{̄}{3}$ $1.5713$ $1.2145$ $1.9415$ $1.2023$ $1.757\stackrel{̄}{4}$ $1.328\stackrel{̄}{3}$ .7 $1.6451$ $1.228\stackrel{̄}{8}$ $1.4565$ $1.3856$ $1.7992$ $1.3717$ $1.628\stackrel{̄}{6}$ $1.515\stackrel{̄}{5}$ .8 $1.4985$ $1.3661$ $1.326\stackrel{̄}{8}$ $1.543\stackrel{̄}{0}$ $1.6389$ $1.527\stackrel{̄}{5}$ $1.483\stackrel{̄}{5}$ $1.6875$ .9 $1.3370$ $1.4917$ $1.183\stackrel{̄}{8}$ $1.6849$ $1.462\stackrel{̄}{3}$ $1.6679$ $1.323\stackrel{̄}{6}$ $1.842\stackrel{̄}{7}$ 1.0 $1.162\stackrel{̄}{2}$ $1.6024$ $1.0289$ $1.8099$ $1.2710$ $1.7917$ $1.150\stackrel{̄}{5}$ $1.979\stackrel{̄}{5}$ 1.1 $0.9756$ $1.6971$ $0.8638$ $1.9168$ $1.067\stackrel{̄}{1}$ $1.8976$ $0.965\stackrel{̄}{9}$ $2.096\stackrel{̄}{5}$ 1.2 $.7794$ $1.774\stackrel{̄}{9}$ $.6900$ $2.0047$ $.8524$ $1.984\stackrel{̄}{6}$ $.771\stackrel{̄}{6}$ $2.1925$ 1.3 $5754$ $1.8349$ $5094$ $2.0725$ $629\stackrel{̄}{3}$ $2.051\stackrel{̄}{7}$ $569\stackrel{̄}{6}$ $2.266\stackrel{̄}{7}$ 1.4 $365\stackrel{̄}{6}$ $1.876\stackrel{̄}{6}$ $323\stackrel{̄}{7}$ $2.1196$ $3998$ $2.0983$ $3619$ $2.318\stackrel{̄}{2}$ 1.5 $152\stackrel{̄}{2}$ $1.8996$ $1347$ $2.1455$ $1664$ $2.1239$ $1506$ $2.3465$ $\frac{1}{2}\pi$ $.0000$ $1.9043$ $0000$ $2.150\stackrel{̄}{9}$ $.0000$ $2.129\stackrel{̄}{3}$ $.0000$ $2.3524$

TABLE III.

 $u$ $gdu$ ${\theta }^{\circ }$ $u$ $gdu$ ${\theta }^{\circ }$ $u$ $gdu$ ${\theta }^{\circ }$ $\circ$ $\circ$ $\circ$ 00 $.0000$ $0.000$ .60 $.5669$ $32.483$ 1.50 $1.1317$ $64.843$ .02 $020\stackrel{̄}{0}$ $1.146$ .62 $583\stackrel{̄}{7}$ $33.444$ 1.55 $1.152\stackrel{̄}{5}$ $66.034$ .04 $040\stackrel{̄}{0}$ $2.291$ .64 $600\stackrel{̄}{3}$ $34.395$ 1.60 $1.172\stackrel{̄}{4}$ $67.171$ .06 $060\stackrel{̄}{0}$ $3.436$ .66 $6167$ $35.336$ 1.65 $1.1913$ $68.257$ .08 $0799$ $4.579$ .68 $6329$ $36.265$ 1.70 $1.2094$ $69.294$ .10 $.0998$ $5.720$ .70 $.6489$ $37.183$ 1.75 $1.226\stackrel{̄}{7}$ $70.284$ .12 $1197$ $6.859$ .72 $6648$ $38.091$ 1.80 $1.243\stackrel{̄}{2}$ $71.228$ .14 $1395$ $7.995$ .74 $6804$ $38.987$ 1.85 $1.258\stackrel{̄}{9}$ $72.128$ .16 $1593$ $9.128$ .76 $6958$ $39.872$ 1.90 $1.273\stackrel{̄}{9}$ $72.987$ .18 $1790$ $10.258$ .78 $7111$ $40.746$ 1.95 $1.2881$ $73.805$ — .20 $.198\stackrel{̄}{7}$ $11.384$ .80 $.7261$ $41.608$ 2.00 $1.3017$ $74.584$ .22 $218\stackrel{̄}{3}$ $12.505$ .82 $7410$ $42.460$ 2.10 $1.3271$ $76.037$ .24 $2377$ $13.621$ .84 $755\stackrel{̄}{7}$ $43.299$ 2.20 $1.350\stackrel{̄}{1}$ $77.354$ .26 $2571$ $14.732$ .86 $770\stackrel{̄}{2}$ $44.128$ 2.30 $1.371\stackrel{̄}{0}$ $78.549$ .28 $2764$ $15.837$ .88 $7844$ $44.944$ 2.40 $1.389\stackrel{̄}{9}$ $79.633$ .30 $.2956$ $16.937$ .90 $.798\stackrel{̄}{5}$ $45.750$ 2.50 $1.407\stackrel{̄}{0}$ $80.615$ .32 $314\stackrel{̄}{7}$ $18.030$ .92 $8123$ $46.544$ 2.60 $1.422\stackrel{̄}{7}$ $81.513$ .34 $3336$ $19.116$ .94 $826\stackrel{̄}{0}$ $47.326$ 2.70 $1.436\stackrel{̄}{6}$ $82.310$ .36 $352\stackrel{̄}{5}$ $20.195$ .96 $8394$ $48.097$ 2.80 $1.4493$ $83.040$ .38 $371\stackrel{̄}{2}$ $21.267$ .98 $8528$ $48.857$ 2.90 $1.460\stackrel{̄}{9}$ $83.707$ — .40 $.3897$ $22.331$ 1.00 $.865\stackrel{̄}{8}$ $49.605$ 3.00 $1.4713$ $84.301$ .42 $408\stackrel{̄}{2}$ $23.386$ 1.05 $897\stackrel{̄}{6}$ $51.428$ 3.10 $1.4808$ $84.841$ .44 $4264$ $24.434$ 1.10 $9281$ $53.178$ 3.20 $1.4894$ $85.336$ .46 $444\stackrel{̄}{6}$ $25.473$ 1.15 $957\stackrel{̄}{5}$ $54.860$ 3.30 $1.497\stackrel{̄}{1}$ $80.715$ .48 $462\stackrel{̄}{6}$ $26.503$ 1.20 $985\stackrel{̄}{7}$ $56.476$ 3.40 $1.504\stackrel{̄}{1}$ $86.177$ .50 $.4804$ $27.524$ 1.25 $1.0127$ $58.026$ 3.50 $1.5104$ $86.541$ .52 $4980$ $28.535$ 1.30 $1.038\stackrel{̄}{7}$ $59.511$ 3.60 $1.516\stackrel{̄}{2}$ $86.870$ .54 $5155$ $29.537$ 1.35 $1.063\stackrel{̄}{5}$ $60.933$ 3.70 $1.5214$ $87.168$ .56 $5328$ $30.529$ 1.40 $1.087\stackrel{̄}{3}$ $62.295$ 3.80 $1.526\stackrel{̄}{1}$ $87.437$ .58 $550\stackrel{̄}{0}$ $31.511$ 1.45 $1.110\stackrel{̄}{0}$ $63.598$ 3.90 $1.5303$ $87.681$

TABLE IV.
 $u$ $gdu$ $logsinhu$ $logcoshu$ $u$ $gdu$ $logsinhu$ $logcoshu$ 4.0 $1.534\stackrel{̄}{2}$ $1.4360$ $1.4363$ 5.5 $1.5626$ $2.08758$ $2.0876\stackrel{̄}{0}$ 4.1 $1.537\stackrel{̄}{7}$ $1.4795$ $1.4797$ 5.6 $1.5634$ $2.13101$ $2.1310\stackrel{̄}{3}$ 4.2 $1.5408$ $1.5229$ $1.5231$ 5.7 $1.5641$ $2.17444$ $2.17445$ 4.3 $1.543\stackrel{̄}{7}$ $1.5664$ $1.5665$ 5.8 $1.5648$ $2.21787$ $2.21788$ 4.4 $1.5462$ $1.6098$ $1.6099$ 5.9 $1.5653$ $2.36130$ $2.26131$ — 4.5 $1.548\stackrel{̄}{6}$ $1.6532$ $1.6533$ 6.0 $1.5658$ $2.30473$ $2.3047\stackrel{̄}{4}$ 4.6 $1.550\stackrel{̄}{7}$ $1.6967$ $1.6968$ 6.2 $1.5667$ $2.39159$ $2.3916\stackrel{̄}{0}$ 4.7 $1.5526$ $1.7401$ $1.7402$ 6.4 $1.567\stackrel{̄}{5}$ $2.47845$ $2.47846$ 4.8 $1.5543$ $1.7836$ $1.7836$ 6.6 $1.568\stackrel{̄}{1}$ $2.56531$ $2.56531$ 4.9 $1.5559$ $1.8270$ $1.8270$ 6.8 $1.568\stackrel{̄}{6}$ $2.65217$ $2.65217$ — 5.0 $1.5573$ $1.8704$ $1.870\stackrel{̄}{5}$ 7.0 $1.569\stackrel{̄}{0}$ $2.73903$ $2.73903$ 5.1 $1.5586$ $1.913\stackrel{̄}{9}$ $1.913\stackrel{̄}{9}$ 7.5 $1.569\stackrel{̄}{7}$ $2.9561\stackrel{̄}{8}$ $3.9561\stackrel{̄}{8}$ 5.2 $1.559\stackrel{̄}{8}$ $1.957\stackrel{̄}{3}$ $1.9573$ 8.0 $1.570\stackrel{̄}{1}$ $3.1733\stackrel{̄}{3}$ $3.1733\stackrel{̄}{3}$ 5.3 $1.5608$ $2.0007$ $2.0007$ 8.5 $1.570\stackrel{̄}{4}$ $3.39047$ $3.39047$ 5.4 $1.561\stackrel{̄}{8}$ $2.044\stackrel{̄}{2}$ $2.044\stackrel{̄}{2}$ 9.0 $1.5705$ $3.60762$ $3.60762$ $\infty$ $1.570\stackrel{̄}{8}$ $\infty$ $\infty$

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