Article 39
Explanation 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

cosh(x ± iy) = a ± ib,sinh(x ± iy) = c ± id,

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 cosh(.82 + 1.34i): First find cosh(.82 + 1.3i), by keeping y at 1.3 and interpolating between the entries under x = .8 and x = .9; next find cosh(.82 + 1.4i), by keeping y at 1.4 and interpolating between the entries under x = .8 and x = .9, as before; then by interpolation between cosh(.82 + 1.3i) and cosh(.82 + 1.4i) find cosh(.82 + 1.34i), 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 §:

sinh(x + 2iπ) = sinhx,cosh(x + 2iπ) = coshx,  etc.

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

cos(y ± ix) = a ib,sin(y ± ix) = d ± ic;

and, moreover, the exponential function is given by

exp(±x ± iy) = a ± c ± i(b ± d),

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 θ = 180v π , 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 ex and ex, 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
sinh u. cosh u. tanh u.
u
sinh u. cosh u. tanh u.








.00.00001.0000.00001.001.17521.5431 ̄.7616
0202001.000202001.021.2063 ̄ 1.5669 ̄ 7699 ̄
0404001.00080400 ̄1.041.2379 ̄1.59137779 ̄
0606001.001805991.061.2700 ̄ 1.61647857 ̄
080801 ̄1.003207981.081.30251.64217932 ̄
.10.1002 ̄1.0050.0997 ̄1.101.33561.6685.8005
121203 ̄ 1.007211941.121.3693 ̄ 1.6956 ̄ 8076 ̄
141405 ̄1.00981391 ̄1.141.4035 ̄1.72338144
161607 ̄ 1.012815861.161.43821.75178210
181810 ̄1.01621781 ̄1.181.47351.78088275 ̄
.20.20131.0201 ̄.1974 ̄1.201.5095 ̄1.8107 ̄.8337 ̄
222218 ̄ 1.0243 ̄ 21651.221.5460 ̄ 1.84128397 ̄
2424231.02892355 ̄1.241.58311.8725 ̄8455 ̄
2626291.0340 ̄ 2543 ̄ 1.261.6209 ̄ 1.90458511 ̄
282837 ̄1.039527291.281.65931.93738565 ̄
.30.30451.0453.29131.301.69841.9709.8617
323255 ̄ 1.051630951.321.73812.0053 ̄ 8668
3434661.0584 ̄3275 ̄1.341.7786 ̄2.04048717 ̄
3636781.065534521.361.8198 ̄ 2.07648764 ̄
3838921.073136271.381.8617 ̄2.11328810 ̄
.40.4108 ̄1.0811 ̄.37991.401.90432.1509 ̄.8854
424325 ̄ 1.089539691.421.94772.18948896 ̄
4445431.0984 ̄41361.441.9919 ̄2.22888937 ̄
464764 ̄ 1.1077 ̄ 4301 ̄ 1.462.0369 ̄ 2.2691 ̄ 8977 ̄
4849861.117444621.482.0827 ̄2.3103 ̄9015 ̄
.50.5211 ̄1.1276.46211.502.1293 ̄2.3524.9051
525438 ̄ 1.1383 ̄ 47771.522.1768 ̄ 2.3955 ̄ 9087 ̄
5456661.1494 ̄4930 ̄1.542.22512.4395 ̄9121
5658971.16095080 ̄ 1.562.27432.4845 ̄ 9154
586131 ̄1.1730 ̄5227 ̄1.582.3245 ̄2.5305 ̄9186
.60.6367 ̄1.1855 ̄.53701.602.3756 ̄2.5775 ̄.9217 ̄
626605 ̄ 1.198455111.622.4276 ̄ 2.6255 ̄ 9246
646846 ̄1.2119 ̄5649 ̄1.642.4806 ̄2.6746 ̄9275 ̄
667090 ̄ 1.22585784 ̄ 1.662.5346 ̄ 2.72479302
6873361.240259151.682.58962.7760 ̄9329 ̄
.70.7586 ̄1.2552 ̄.6044 ̄1.702.64562.8283.9354
7278381.2706 ̄ 61691.722.70272.8818 ̄ 9379 ̄
7480941.286562911.742.76092.93649402
7683531.3030 ̄ 6411 ̄ 1.762.8202 ̄ 2.99229425
7886151.319965271.782.88063.04929447 ̄
.80.88811.3374.66401.802.9422 ̄3.1075 ̄.9468
8291501.3555 ̄ 6751 ̄ 1.823.00493.16699488
8494231.374068581.843.0689 ̄3.2277 ̄9508 ̄
869700 ̄ 1.3932 ̄ 6963 ̄ 1.863.13403.28979527 ̄
889981 ̄1.412870641.883.2005 ̄3.35309545 ̄
.901.02651.4331 ̄.7163 ̄1.903.2682 ̄3.4177.9562
921.05541.45397259 ̄ 1.923.3372 ̄ 3.4838 ̄ 9579
941.0847 ̄1.475373521.943.40753.55129595
961.11441.4973 ̄ 7443 ̄ 1.963.47923.6201 ̄ 9611 ̄
981.1446 ̄1.5199 ̄7531 ̄1.983.55233.69049626 ̄
2.003.6269 ̄3.7622 ̄.96403.0010.0179 ̄10.0677 ̄.99505
2.023.70283.8355 ̄96543.0210.221210.270099524
2.043.7803 ̄3.910396673.0410.428710.476599543
2.063.8593 ̄3.986796803.0610.640310.687299561
2.083.93984.06479693 ̄3.0810.856210.9022 ̄99578
2.104.0219 ̄4.1443.9705 ̄3.1011.0765 ̄11.1215.99594
2.124.10554.2256 ̄9716 ̄3.1211.301111.3453 ̄99610
2.144.1909 ̄4.30859727 ̄3.1411.5303 ̄11.5736 ̄99626
2.164.27794.393297373.1611.7641 ̄11.806599640
2.184.36664.4797 ̄9748 ̄3.1812.0026 ̄12.0442 ̄99654
2.204.45714.5679.97573.2012.2459 ̄12.2866.99668
2.224.5494 ̄4.6580 ̄9767 ̄3.2212.4941 ̄12.534099681
2.244.64344.7499 ̄9776 ̄3.2412.7473 ̄12.786499693
2.264.7394 ̄4.84379785 ̄3.2613.0056 ̄13.0440 ̄99705
2.284.8372 ̄4.9395 ̄9793 ̄3.2813.2691 ̄13.306799717
2.304.9370 ̄5.0372.9801 ̄3.3013.5379 ̄13.5748 ̄.99728
2.325.03875.1370 ̄9809 ̄3.3213.8121 ̄13.8483 ̄99738
2.345.14255.2388 ̄98163.3414.091814.1273 ̄99749
2.365.2483 ̄5.3427 ̄98233.3614.377214.4120 ̄99758
2.385.3562 ̄5.448798303.3814.6684 ̄14.702499768
2.405.46625.5569.9837 ̄3.4014.9654 ̄14.9987.99777
2.425.57855.6674 ̄98433.4215.2684 ̄15.3011 ̄99786
2.445.69295.780198493.4415.577415.609599794
2.465.8097 ̄5.895198553.4615.8928 ̄15.924299802
2.485.9288 ̄6.01259861 ̄3.4816.214416.2453 ̄99810
2.506.05026.1323 ̄.98663.5016.542616.5728.99817
2.526.1741 ̄6.254598713.5216.877416.907099824
2.546.30046.3793 ̄98763.5417.2190 ̄17.2480 ̄99831
2.566.4293 ̄6.5066 ̄98813.5617.5674 ̄17.595899831
2.586.5607 ̄6.63649886 ̄3.5817.922817.950799844
2.606.69476.7690.98903.6018.285418.3128 ̄.99850
2.626.8315 ̄6.9043 ̄9895 ̄3.6218.6554 ̄18.6822 ̄99856
2.646.97097.0423 ̄9899 ̄3.6419.0328 ̄19.059099862
2.667.1132 ̄7.1832 ̄9903 ̄3.6619.417819.443599867
2.687.2583 ̄7.326899063.6819.8106 ̄19.835899872
2.707.4063 ̄7.4735 ̄.99103.7020.2113 ̄20.2360.99877
2.727.55727.6231 ̄9914 ̄3.7220.6201 ̄20.644399882
2.747.71127.7758 ̄9917 ̄3.7421.037121.0609 ̄99887
2.767.8683 ̄7.9316 ̄99203.7621.4626 ̄21.4859 ̄99891
2.788.0285 ̄8.090599233.7821.896621.919499896 ̄
2.808.19198.2527.99263.8022.339422.3618 ̄.99900 ̄
2.828.35868.418299293.8222.791122.8131 ̄99904 ̄
2.848.5287 ̄8.5871 ̄9932 ̄3.8423.2520 ̄23.2735 ̄99907
2.868.70218.7594 ̄9935 ̄3.8623.722123.743299911
2.888.8791 ̄8.935299373.8824.201824.222499915 ̄
2.909.0596 ̄9.1146 ̄.9940 ̄3.9024.691124.7113.99918
2.929.2437 ̄9.29769942 ̄3.9225.190325.210199921
2.949.4315 ̄9.4844 ̄99443.9425.6996 ̄25.719099924
2.969.6231 ̄9.6749 ̄9947 ̄3.9626.219126.2382 ̄99927
2.989.81859.86939949 ̄3.9826.7492 ̄26.7679 ̄99930








TABLE II.—VALUES OF cosh(x + iy) AND sinh(x + iy).










x = 0
x = .1








y
a
b
c
d
a
b
c
d









01.000000000000.00001.0050.00000.10017 ̄ .0000
.10.995009981.0000 ̄01000099671003
.20.9801 ̄ 1987 ̄ 0.985001990 ̄ 098171997 ̄
.30.955329550.96010296009570 ̄2970 ̄
.4.9211 ̄.3894.9257 ̄.03901.09226.3914
.5877647948820 ̄ 0480208791 ̄ 4818
.6825356468295 ̄05656082675675 ̄
.7764864427687 ̄ 06453076616474
.8.6967.7174 ̄.7002 ̄.07186 ̄.06979 ̄.7800
.9621678336247 ̄07847 ̄06227 ̄7872
1.054038415 ̄543008429054128457 ̄
1.1453689124559 ̄08927045448957 ̄
1.2.3624 ̄.9320.3642 ̄.09336.03630 ̄0.9367 ̄
1.326759636 ̄2688 ̄09652 ̄02680 ̄0.9684 ̄
1.41700 ̄985417080987101703 ̄0.9904 ̄
1.507079975 ̄071109992 ̄00709 ̄1.0025 ̄
1 2π00001.0000000010017 ̄000001.0050



















x = .2
x = .3








y
a
b
c
d
a
b
c
d









01.0201 ̄ .0000.2013.00001.0453.0000.3045.0000
.11.0150 ̄0201200310181.0401 ̄03043030 ̄1044
.20.9997040019732027 ̄ 1.0245 ̄ 06052985 ̄ 2077 ̄
.30.974505951923301499870900 ̄29093089
.4.9395.0784.1854.3972.9628.1186.2805 ̄.4071 ̄
.58952 ̄ 09651767 ̄ 48909174 ̄ 1460 ̄ 2672 ̄ 5012 ̄
.684191137 ̄1662 ̄5760 ̄8687171925135903 ̄
.77802 ̄ 12971540 ̄ 657179951962 ̄ 23296734
.8.7107 ̄.1444.1403 ̄.7318 ̄.7283 ̄.2184.2122 ̄.7498
.96341 ̄15771252 ̄79906498 ̄23851893 ̄8188
1.0551116941088 ̄8584 ̄5648256216458796
1.146271795 ̄09139091 ̄4742 ̄271413819316
1.2.3696.1877 ̄.0730 ̄0.9507.3788 ̄.2838.11030.9743 ̄
1.32729 ̄19400539 ̄0.9829 ̄279629340815 ̄1.0072
1.41734 ̄198403421.00521777 ̄30010518 ̄1.0301
1.50722 ̄200801421.017507393038 ̄02151.0427 ̄
1 2π0000201300001.0201 ̄0000304500001.0453









TABLE II.—VALUES OF cosh(x + iy) AND sinh(x + iy). (continued)










x = .4
x = .5








y
a
b
c
d
a
b
c
d









01.0811 ̄ .0000.4108 ̄ .00001.1276.0000.5211 ̄ .0000
.11.075604104087 ̄10791.1220 ̄05205185 ̄1126
.21.059508164026 ̄ 2148 ̄ 1.1051102551072240
.31.0328 ̄1214 ̄39243195 ̄1.0773 ̄1540 ̄49783332
.4.9957.1600 ̄.3783.4210 ̄1.0386.2029.4800 ̄.4391
.5948719693605 ̄ 5183 ̄ 0.9896 ̄ 249845735406
.689222319339061040.930629424301 ̄6367
.7826826463142 ̄ 69640.86243357 ̄ 3986 ̄ 7264
.8.7532 ̄.2947.2862 ̄.7755.7856.3738.3631 ̄0.8089
.96720 ̄32182553846870094082 ̄32390.8833
1.05841345622199097 ̄6093 ̄4385 ̄28150.9489 ̄
1.149043661 ̄18639635 ̄5115 ̄46442364 ̄1.0050 ̄
1.2.3917.3289 ̄.14881.0076.4056.4857 ̄.18881.0510 ̄
1.32892 ̄3958 ̄1099 ̄1.0417 ̄301650211394 ̄1.0865
1.41838 ̄4048 ̄06981.06531917 ̄51350886 ̄1.1163
1.50765 ̄40970291 ̄1.0784 ̄0798 ̄5198 ̄0369 ̄1.1248 ̄
1 2π00004108 ̄00001.0811 ̄00005211 ̄00001.1276


















x = .6
x = .7








y
a
b
c
d
a
b
c
d









01.1855 ̄ .0000.6367 ̄ .00001.2552.0000.7586 ̄ .0000
.11.17950636 ̄6335 ̄11831.2489 ̄07577548 ̄1253
.21.1618 ̄ 1265 ̄ 6240 ̄ 23551.230115427435 ̄ 2494 ̄
.31.1325 ̄1881608235031.19912242 ̄72473709
.41.0918.2479.5864.4617 ̄1.1561 ̄.2954.6987.4888 ̄
.51.04033052558756841.10153637 ̄ 66576018 ̄
.60.97843955 ̄5255 ̄6694 ̄1.035942536261 ̄7087
.70.9067 ̄ 410148697637 ̄ 0.9600 ̄ 4887 ̄ 5802 ̄ 8086
.8.8259.4567.4436 ̄0.8504.8745 ̄.5442 ̄.52850.9004
.97369 ̄498739570.92867802594247150.9832
1.0640553573440 ̄0.99756782 ̄63834099 ̄1.0562 ̄
1.153775674 ̄2888 ̄1.0565 ̄569367603441 ̄1.1186
1.2.4296 ̄.5934 ̄.2307 ̄1.1049 ̄.4548.7070.2749 ̄1.1699 ̄
1.331716135 ̄17031.14223358 ̄730920291.2094
1.42015 ̄6274 ̄10821.16822133747512891.2369
1.50839 ̄6351 ̄04501.1825 ̄0888 ̄7567 ̄0537 ̄1.2520
1 2π00006367 ̄00001.1855 ̄0000758600001.2552









TABLE II.—VALUES OF cosh(x + iy) AND sinh(x + iy). (continued)










x = .8
x = .9








y
a
b
c
d
a
b
c
d









01.3374.0000.8881.00001.4331 ̄ .00001.0265.0000
.11.3308 ̄0887 ̄8837 ̄13351.42591025 ̄1.0214 ̄1431 ̄
.21.31081764870426571.404520391.0061 ̄ 2847
.31.27762625 ̄848439521.36913034 ̄0.9807 ̄4235
.41.2319 ̄.3458.8180.52081.3200 ̄.3997.9455 ̄.5581 ̄
.51.1737 ̄ 4258 ̄ 7794 ̄ 6412 ̄ 1.2577 ̄ 492190086871 ̄
.61.10385015 ̄7330 ̄7552 ̄1.1828 ̄579684728092 ̄
.71.022957216793 ̄ 8616 ̄ 1.0961 ̄ 6613 ̄ 78519232
.8.9318 ̄.6371 ̄.6188 ̄0.9595.9984.7364 ̄.7152 ̄1.0280
.98314 ̄6957 ̄5521 ̄1.047689088041 ̄6381 ̄1.1226
1.07226747247981.12547743863855461.2059 ̄
1.16067 ̄7915 ̄40281.19196500914846561.2772 ̄
1.2.4846.8278 ̄.32181.2465.5193 ̄0.9568 ̄.3720 ̄1.3357 ̄
1.33578 ̄85572376 ̄1.2887 ̄3834 ̄0.98912746 ̄1.3809 ̄
1.422738752 ̄1510 ̄1.318024361.012417451.4122
1.509468859 ̄06281.3341 ̄1014 ̄1.023907261.4295 ̄
1 2π0000.888100001.337400001.026500001.4331 ̄


















x = 1.0
x = 1.1








y
a
b
c
d
a
b
c
d









01.5431 ̄ .00001.1752.00001.6685.00001.3356.0000
.11.5354 ̄11731.16931541 ̄1.6602 ̄13331.3290 ̄1666
.21.512323351.15183066 ̄ 1.6353 ̄ 26541.30903315 ̄
.31.4742 ̄3473 ̄1.122745601.5940 ̄39461.2760 ̄4931 ̄
.41.4213 ̄4576 ̄1.0824.60091.536852011.23020.6498 ̄
.51.3542 ̄ 56341.0314 ̄ 7398 ̄ 1.4643 ̄ 64031.17210.7999
.61.2736 ̄6636 ̄0.96998718 ̄1.3771 ̄7542 ̄1.1024 ̄0.9421
.71.18027571 ̄ 0.89889941 ̄ 1.2762 ̄ 86041.0216 ̄ 1.0749 ̄
.81.0751 ̄0.8430.8188 ̄1.10691.1625 ̄0.9581.9306 ̄1.1969
.90.95920.9206 ̄73051.20871.0372 ̄1.046283021.3070
1.00.83370.98896350 ̄1.2985 ̄0.90151.12397217 ̄1.4040
1.10.69991.04735331 ̄1.3752 ̄0.75681.190360581.4870 ̄
1.2.5592 ̄1.0953.42581.4382.60461.2449 ̄.4840 ̄1.5551
1.351281.1324 ̄3144 ̄1.4868 ̄44631.2870 ̄3575 ̄1.6077
1.42623 ̄1.1581 ̄1998 ̄1.521328361.316222701.6442
1.51092 ̄1.1723 ̄08311.539211801.3323 ̄0945 ̄1.6643
1 2π00001.175200001.5431 ̄.00001.3356.00001.6685









TABLE II.—VALUES OF cosh(x + iy) AND sinh(x + iy). (continued)










x = 1.2
x = 1.3








y
a
b
c
d
a
b
c
d









01.8107 ̄ .00001.5095 ̄ .00001.970900001.6984 ̄ .0000
.11.80161507 ̄1.50191808 ̄1.9611 ̄1696 ̄1.6899 ̄1968 ̄
.21.7746 ̄ 2999 ̄ 1.4794 ̄ 3598 ̄ 1.931633741.66453916
.31.7298 ̄4461 ̄1.44205351 ̄1.8829 ̄50191.62255824
.41.6677.58781.39030.70511.8153.6614 ̄1.56430.7675
.51.58907237 ̄ 1.3247 ̄ 0.8681 ̄ 1.729681421.4905 ̄ 0.9449
.61.494485231.24581.0224 ̄1.6267 ̄9590 ̄1.40171.1131
.71.3849 ̄ 97241.1545 ̄ 1.1665 ̄ 1.50741.09411.2990 ̄ 1.2697
.81.2615 ̄1.08281.0517 ̄1.2989 ̄1.37311.21831.1833 ̄1.4139 ̄
.91.12551.1824 ̄0.9383 ̄1.41831.22511.3304 ̄1.05571.5439 ̄
1.00.97831.2702 ̄0.8156 ̄1.52361.0649 ̄1.42910.91761.6585 ̄
1.10.82131.34520.6847 ̄1.6137 ̄0.89401.51360.7704 ̄1.7565 ̄
1.2.65611.4069 ̄0.5470 ̄1.6876.7142 ̄1.5830 ̄0.61541.8370 ̄
1.34844 ̄1.45440.4038 ̄1.7447 ̄52721.6365 ̄0.45431.8991 ̄
1.43078 ̄1.4875 ̄0.2566 ̄1.784333501.6737 ̄0.2887 ̄1.9422
1.51281 ̄1.5057 ̄0.1068 ̄1.806113941.69410.12011.9660 ̄
1 2π00001.5095 ̄00001.8107 ̄00001.6984 ̄00001.9709


















x = 1.4
x = 1.5








y
a
b
c
d
a
b
c
d









02.1509 ̄ .00001.9043.00002.3524.00002.1293 ̄ .0000
.12.140119011.894821472.341321262.1187 ̄2348
.22.108037831.866342732.305542302.08684674
.32.05485628 ̄1.819263562.247362922.0342 ̄6951
.41.98110.7416 ̄1.75400.83762.16670.8292 ̄1.9612 ̄0.9161 ̄
.51.8876 ̄ 0.9130 ̄ 1.6712 ̄ 1.0312 ̄ 2.06441.02081.86861.1278
.61.77521.0753 ̄1.57131.21451.94151.20231.7574 ̄1.3283 ̄
.71.64511.2288 ̄ 1.45651.38561.79921.37171.6286 ̄ 1.5155 ̄
.81.49851.36611.3268 ̄1.5430 ̄1.63891.5275 ̄1.4835 ̄1.6875
.91.33701.49171.1838 ̄1.68491.4623 ̄1.66791.3236 ̄1.8427 ̄
1.01.1622 ̄1.60241.02891.80991.27101.79171.1505 ̄1.9795 ̄
1.10.97561.69710.86381.91681.0671 ̄1.89760.9659 ̄2.0965 ̄
1.2.77941.7749 ̄.69002.0047.85241.9846 ̄.7716 ̄2.1925
1.357541.834950942.07256293 ̄2.0517 ̄5696 ̄2.2667 ̄
1.43656 ̄1.8766 ̄3237 ̄2.119639982.098336192.3182 ̄
1.51522 ̄1.899613472.145516642.123915062.3465
1 2π.00001.904300002.1509 ̄.00002.1293 ̄.00002.3524









TABLE III.










u
gd u
θ
u
gd u
θ
u
gd u
θ









00.00000.000 .60.566932.483 1.501.131764.843
.020200 ̄1.146 .625837 ̄33.444 1.551.1525 ̄66.034
.040400 ̄ 2.291 .646003 ̄ 34.395 1.601.1724 ̄ 67.171
.060600 ̄3.436 .66616735.336 1.651.191368.257
.0807994.579 .68632936.265 1.701.209469.294
.10.09985.720 .70.648937.183 1.751.2267 ̄ 70.284
.1211976.859 .72664838.091 1.801.2432 ̄71.228
.1413957.995 .74680438.987 1.851.2589 ̄ 72.128
.1615939.128 .76695839.872 1.901.2739 ̄72.987
.18179010.258 .78711140.746 1.951.288173.805
.20.1987 ̄ 11.384 .80.726141.608 2.001.301774.584
.222183 ̄12.505 .82741042.460 2.101.327176.037
.24237713.621 .847557 ̄ 43.299 2.201.3501 ̄ 77.354
.26257114.732 .867702 ̄44.128 2.301.3710 ̄78.549
.28276415.837 .88784444.944 2.401.3899 ̄ 79.633
.30.295616.937 .90.7985 ̄45.750 2.501.4070 ̄80.615
.323147 ̄18.030 .92812346.544 2.601.4227 ̄81.513
.34333619.116 .948260 ̄47.326 2.701.4366 ̄82.310
.363525 ̄20.195 .96839448.097 2.801.449383.040
.383712 ̄21.267 .98852848.857 2.901.4609 ̄83.707
.40.389722.331 1.00.8658 ̄49.605 3.001.471384.301
.424082 ̄23.386 1.058976 ̄51.428 3.101.480884.841
.44426424.434 1.10928153.178 3.201.489485.336
.464446 ̄25.473 1.159575 ̄54.860 3.301.4971 ̄80.715
.484626 ̄26.503 1.209857 ̄56.476 3.401.5041 ̄86.177
.50.480427.524 1.251.012758.026 3.501.510486.541
.52498028.535 1.301.0387 ̄59.511 3.601.5162 ̄86.870
.54515529.537 1.351.0635 ̄60.933 3.701.521487.168
.56532830.529 1.401.0873 ̄62.295 3.801.5261 ̄87.437
.585500 ̄31.511 1.451.1100 ̄63.598 3.901.530387.681










TABLE IV.








u
gd u
log sinh u
log cosh u
u
gd u
log sinh u
log cosh u








4.01.5342 ̄ 1.43601.4363 5.51.56262.087582.08760 ̄
4.11.5377 ̄1.47951.4797 5.61.56342.131012.13103 ̄
4.21.54081.52291.5231 5.71.56412.174442.17445
4.31.5437 ̄1.56641.5665 5.81.56482.217872.21788
4.41.54621.60981.6099 5.91.56532.361302.26131
4.51.5486 ̄ 1.65321.6533 6.01.56582.304732.30474 ̄
4.61.5507 ̄1.69671.6968 6.21.56672.391592.39160 ̄
4.71.55261.74011.7402 6.41.5675 ̄2.478452.47846
4.81.55431.78361.7836 6.61.5681 ̄2.565312.56531
4.91.55591.82701.8270 6.81.5686 ̄2.652172.65217
5.01.55731.87041.8705 ̄ 7.01.5690 ̄2.739032.73903
5.11.55861.9139 ̄1.9139 ̄ 7.51.5697 ̄2.95618 ̄3.95618 ̄
5.21.5598 ̄1.9573 ̄1.9573 8.01.5701 ̄3.17333 ̄3.17333 ̄
5.31.56082.00072.0007 8.51.5704 ̄3.390473.39047
5.41.5618 ̄2.0442 ̄2.0442 ̄ 9.01.57053.607623.60762
1.5708 ̄