II
Quaternion Theorems

1. Definitions

1. As there are two or three symbols and terms which will be in constant use in the following pages that are new or more general in their signification than is usual, it is necessary to be perhaps somewhat tediously minute in a few preliminary definitions and explanations.

A function of a variable in the following essay is to be understood to mean anything which depends on the variable and which can be subjected to mathematical operations; the variable itself being anything capable of being represented by a mathematical symbol. In Cartesian Geometry the variable is generally a single scalar. In Quaternions on the other hand a general quaternion variable is not infrequent, a variable which requires 4 scalars for its specification, and similarly for the function. In both, however, either the variable or the function may be a mere symbol of operation. In the following essay we shall frequently have to speak of variables and functions which are neither quaternions nor mere symbols of operation. For instance $K$ in Section IV below requires 6 scalars to specify it, and it is a function of $\psi$ which requires 6 scalars and $\rho$ which requires 3 scalars. When in future the expression “any function” is used it is always to be understood in the general sense just explained.

We shall frequently have to deal with functions of many independent vectors, and especially with functions which are linear in each of the constituent vectors. These functions merely require to be noticed but not defined.

Hamilton has defined the meaning of the symbolic vector $\nabla$ thus:—

where $i$, $j$, $k$ are unit vectors in the directions of the mutually perpendicular axes $x$, $y$, $z$. I have found it necessary somewhat to expand the meaning of this symbol. When a numerical suffix $1$, $2$, … is attached to a $\nabla$ in any expression it is to indicate that the differentiations implied in the $\nabla$ are to refer to and only to other symbols in the same expression which have the same suffix. After the implied differentiations have been performed the suffixes are of course removed. Thus being a quaternion function of any four vectors $\alpha$, $\beta$, $\gamma$, $\delta$, linear in each

and again

It is convenient to reserve the symbol $\Delta$ for a special meaning. It is to be regarded as a particular form of $\nabla$, but its differentiations are to refer to all the variables in the term in which it appears. Thus $Q$ being as before

If in a linear expression or function ${\nabla }_1$ and ${\rho }_1$ ($\rho$ being as usual $\equiv ix+jy+kz$) occur once each they can be interchanged. Similarly for ${\nabla }_2$ and ${\rho }_2$. So often does this occur that I have thought it advisable to use a separate symbol ${\zeta }_1$ for each of the two ${\nabla }_1$ and ${\rho }_1$, ${\zeta }_2$ for each of the two ${\nabla }_2$ and ${\rho }_2$ and so for ${\zeta }_3$, &c. If only one such pair occur there is of course no need for the suffix attached to $\zeta$. Thus $\zeta$ may be looked upon as a symbolic vector or as a single term put down instead of three. For being linear in each of the vectors $\alpha$, $\beta$

There is one more extension of the meaning of $\nabla$ to be given. $u$, $v$, $w$ being the rectangular coordinates of any vector $\sigma$, $\sigma \nabla$ is defined by the equation

To $\sigma \nabla$ of course are to be attached, when necessary, the suffixes above explained in connection with $\nabla$. Moreover just as for ${\nabla }_1$, ${\rho }_1$ we may put $\zeta$, $\zeta$ so also for $\sigma {\nabla }_1$, ${\sigma }_1$ may we put the same.

With these meanings one important result follows at once. The ${\nabla }_1$’s, ${\nabla }_2$’s, &c., obey all the laws of ordinary vectors whether with regard to multiplication or addition, for the coordinates $d∕dx$, $d∕dy$, $d∕dz$ of any $\nabla$ obey with the coordinates of any vector or any other $\nabla$ all the laws of common algebra.

Just as $\sigma \nabla$ may be defined as a symbolic vector whose coordinates are $d∕du$, $d∕dv$, $d∕dw$ so $\varphi$ being a linear vector function of any vector whose coordinates are

$\varphi D$⁶ is defined as a symbolic linear vector function whose coordinates are

and to $\varphi D$ is to be applied exactly the same system of suffixes as in the case of $\nabla$. Thus $q$ being any quaternion function of $\varphi$, and $\omega$ any vector

The same symbol $\varphi D$ is used without any inconvenience with a slightly different meaning. If the independent variable $\varphi$ be a self-conjugate linear vector function it has only six coordinates. If these are $PQRSTU$ (i.e. $\varphi i=Pi+Uj+Tk$, &c.) $\varphi D$ is defined as a self-conjugate linear vector function whose coordinates are

We shall frequently have to compare volume integrals with integrals taken over the bounding surface of the volume, and again surface integrals with integrals taken round the boundary of the surface. For this purpose we shall use the following notations for linear, surface and volume integrals respectively $\int \; <!--nolimits-->Qd\rho$, $\iint Qd\Sigma$, $\iiint Qd\mathfrak{𝔰}$ where $Q$ is any function of the position of a point. Here $d\rho$ is a vector element of the curve, $d\Sigma$ a vector element of the surface, and $d\mathfrak{𝔰}$ an element of volume. When comparisons between line and surface integrals are made we take $d\Sigma$ in such a direction that $d\rho$ is in the direction of positive rotation round the element $d\Sigma$ close to it. When comparisons between surface and volume integrals are made $d\Sigma$ is always taken in the direction away from the volume which it bounds.

2. Properties of $\zeta$

2. The property of $\zeta$ on which nearly all its usefulness depends is that if $\sigma$ be any vector

which is given at once by equation (1) of last section.

This gives a useful expression for the conjugate of a linear vector function of a vector. Let $\varphi$ be the function and $\omega$, $\tau$ any two vectors. Then $\varphi \prime$ denoting as usual the conjugate of $\varphi$ we have

whence putting on the left $\tau =-\zeta S\zeta \tau$ we have

or since $\tau$ is quite arbitrary

From this we at once deduce expressions for the pure part $\overline{\varphi }\omega$ and the rotational part $V𝜖\omega$ of $\varphi \omega$ by putting

And all the other well-known relations between $\varphi$ and $\varphi \prime$ are at once given e.g. $S\zeta \varphi \zeta =S\zeta \varphi \prime \zeta$, i.e. the “convergence” of $\varphi =$ the “convergence” of $\varphi \prime$.

3. Let be any function of two vectors which is linear in each. Then if $\varphi \omega$ be any linear vector function of a vector $\omega$ given by

or more generally

To prove, it is only necessary to observe that

As a particular case of eq. (4) let $\varphi$ have the self-conjugate value

or if is symmetrical in $\lambda$ and $\mu$

The application we shall frequently make of this is to the case when for $\alpha$ we put ${\nabla }_1$ and for $\beta$, ${\sigma }_1$, where $\sigma$ is any vector function of the position of a point. In this case the first expression for $\varphi$ is the strain function and the second expression the pure strain function resulting from a small displacement $\sigma$ at every point. As a simple particular case put so that $Q$ is symmetrical in $\lambda$ and $\mu$. Thus $\varphi$ being either of these functions

Another important equation is

This is quite independent of the form of $\varphi$. To prove, observe that by equation (2)

and that $-\zeta S\zeta \varphi {\zeta }_1=\varphi {\zeta }_1$. Thus we get rid of $\zeta$ and may now drop the suffix of ${\zeta }_1$ and so get eq. (6$a$). [Notice that by means of (6$a$), (4$a$) may be deduced from (4); for by (6$a$)

3a. A more important result is the expression for ${\varphi }^{-1}\omega$ in terms of $\varphi$. We assume that

where $\lambda$, $\mu$, $\nu$ are any three vectors and $m$ is a scalar independent of these vectors. Substituting ${\zeta }_1$, ${\zeta }_2$, ${\zeta }_3$ for $\lambda$, $\mu$, $\nu$ and multiplying by $S{\zeta }_1{\zeta }_2{\zeta }_3$

which gives $m$ in terms of $\varphi$. That $S{\zeta }_1{\zeta }_2{\zeta }_{3}S{\zeta }_1{\zeta }_2{\zeta }_3=6$ is seen by getting rid of each pair of $\zeta$’s in succession thus:—

Next observe that

Multiplying by $V{\zeta }_1{\zeta }_2$ and again on the right getting rid of the $\zeta$s we have

whence from equation (6$b$)

or changing $\omega$ into ${\varphi }^{-1}\omega$

By equation (6$a$) of last section we can also put this in the form

so that ${\varphi }^{-1}\omega$ is obtained explicitly in terms of $\varphi$ or $\varphi \prime$.

Equation (6$c$) or (6$d$) can be put in another useful form which is more analogous to the ordinary cubic and can be easily deduced therefrom, or⁸less easily from (6$d$), viz. 

As a useful particular case of equation (6$d$) we may notice that by equation (4) Section 2 if

4. Let $\varphi$, $\psi$ be two linear vector functions of a vector. Then if

where $\chi$ is a quite arbitrary linear vector function

for we may put $\chi \zeta =\tau S\zeta \omega$ where $\tau$ and $\omega$ are arbitrary vectors, so that

Similarly⁹ if $\varphi$ and $\psi$ are both self-conjugate and $\chi$ is a quite arbitrary self-conjugate linear vector function the same relation holds as can be seen by putting

3. Fundamental Property of $D$

5. Just as the fundamental property of $\sigma \nabla$ is that, $Q$ being any function of $\sigma$

so we have a similar property of $D$. $Q$ being any function of $\varphi$ a linear vector function

The property is proved in the same way as for $\nabla$, viz. by expanding $S\delta \varphi \zeta \varphi D_1\zeta$ in terms of the coordinates of $\varphi D$. First let $\varphi$ be not self-conjugate, and let its nine coordinates be

Thus

The proposition is exactly similarly proved when $\varphi$ is self-conjugate.

4. Theorems in Integration

6. Referring back to Section 1 above for our notation for linear surface and volume integrals we will now prove that if $Q$ be any linear function of a vector¹⁰

To prove the first divide the surface up into a series of elementary parallelograms by two families of lines—one or more members of one family coinciding with the given boundary,—apply the line integral to the boundary of each parallelogram and sum for the whole. The result will be the linear integral given in equation (8). Let the sides of one such parallelogram taken in order in the positive direction be $\alpha$, $\beta +\beta \prime$, $-\alpha -\alpha \prime$, $-\beta$; so that $\alpha \prime$ and $\beta \prime$ are infinitely small compared with $\alpha$ and $\beta$, and we have the identical relation

The terms contributed to $\int \; <!--nolimits-->Qd\rho$ by the sides $\alpha$ and $-\alpha -\alpha \prime$ will be (neglecting terms of the third and higher orders of small quantities)

Similarly the terms given by the other two sides will be

so that remembering that $\beta \prime -\alpha \prime =0$ and therefore $Q\beta \prime -Q\alpha \prime =0$ we have for the whole boundary of the parallelogram

where $d\Sigma$ is put for $V\alpha \beta$. Adding for the whole surface we get equation (8).

Equation (9) is proved in an exactly similar way by splitting the volume up into elementary parallelepipeda by three families of surfaces one or more members of one of the families coinciding with the given boundary. If $\alpha$, $\beta$, $\gamma$ be the vector edges of one such parallelepiped we get a term corresponding to $Q\beta \prime -Q\alpha \prime$ viz. 

and we get the sum of three terms corresponding to

above, viz. 

whence putting $S\alpha \beta \gamma =-d\mathfrak{𝔰}$ we get equation (9).

7. It will be observed that the above theorems have been proved only for cases where we can put $dQ=-Q_{1}Sd\rho {\nabla }_1$ i.e. when the space fluxes of $Q$ are finite. If at any isolated point they are not finite this point must be shut off from the rest of the space by a small closed surface or curve as the case may be and this surface or curve must be reckoned as part of the boundary of the space. If at a surface (or curve) $Q$ has a discontinuous value so that its derivatives are there infinite whereas on each side they are finite, this surface (or curve) must be considered as part of the boundary and each element of it will occur twice, i.e. once for the part of the space on each side.

In the case of the isolated points, if the surface integral or line integral round this added boundary vanish, we can of course cease to consider these points as singular. Suppose $Q$ becomes infinite at the point $\rho =\alpha$. Draw a small sphere of radius $a$ and also a sphere of unit radius with the point $\alpha$ for centre, and consider the small sphere to be the added boundary. Let $d\Sigma \prime$ be the element of the unit sphere cut off by the cone which has $\alpha$ for vertex and the element $d\Sigma$ of the small sphere for base. Then $d\Sigma =a^{2}d\Sigma \prime$ and we get for the part of the surface integral considered $a^2\iint Q_{d\Sigma }d\Sigma \prime$ where $Q_{d\Sigma }$ is the value of $Q$ at the element $d\Sigma$. If then

the point may be regarded as not singular. If the limiting expression is finite the added surface integral will be finite. If the expression is infinite the added surface integral will be generally but not always infinite. Similarly in the case of an added line integral if is zero or finite, the added line integral will be zero or finite respectively (of course including in the term finite a possibility of zero value). If this expression be infinite, the added line integral will generally also be infinite.

This leads to the consideration of potentials which is given in Section 5.

8. Some particular cases of equations (8) and (9) which (except the last) have been proved by Tait, are very useful. First put $Q=$ a simple scalar $P$. Thus

If $P$ be the pressure in a fluid $-\iint Pd\Sigma$ is the force resulting from the pressure on any portion and equation (11) shews that $-\nabla P$ is the force per unit volume due to the same cause. Next put $Q\omega =S\omega \sigma$ and $V\omega \sigma$. Thus

Equations (12) and (14) are well-known theorems, and (13) and (15) will receive applications in the following pages. Green’s Theorem with Thomson’s extension of it are, as indeed has been pointed out by Tait particular cases of these equations.

Equations (14) and (15) applied to an element give the well-known physical meanings for $S\nabla \sigma$ and $V\nabla \sigma$¹¹. The first is obtained by applying (14) to any element, and the second (regarding $\sigma$ as a velocity) is obtained by applying (15) to the element contained by the following six planes each passing infinitely near to the point considered—(1) two planes containing the instantaneous axis of rotation, (2) two planes at right angles to this axis, and (3) two planes at right angles to these four.

One very frequent application of equation (9) may be put in the following form:—$Q$ being any linear function (varying from point to point) of $R_1$ and ${\nabla }_1$, $R$ being a function of the position of a point

5. Potentials

9. We proceed at once to the application of these theorems in integration to Potentials. Although the results about to be obtained are well-known ones in Cartesian Geometry or are easily deduced from such results it is well to give this quaternion method if only for the collateral considerations which on account of their many applications in what follows it is expedient to place in this preliminary section.

If $R$ is some function of $\rho -\rho \prime$ where $\rho \prime$ is the vector coordinate of some point under consideration and $\rho$ the vector coordinate of any point in space, we have

Now let be any function of $R$, the coordinates of $Q$ being functions of $\rho$ only. Consider the integral the variable of integration being $\rho$ ($\rho \prime$ being a constant so far as the integral is concerned). It does not matter whether the integral is a volume, surface or linear one but for conciseness let us take it as a volume integral. Thus we have

Now $\rho \nabla$ operating on the whole integral has no meaning so we may drop the affix to the $\nabla$ outside and always understand $\rho \prime$. Under the integral sign however we must retain the affix $\rho$ or $\rho \prime$ unless a convention be adopted. It is convenient to adopt such a convention and since $Q$ will probably contain some $\rho \nabla$ but cannot possibly contain a $\rho \prime \nabla$ we must assume that when $\nabla$ appears without an affix under the integral sign the affix $\rho$ is understood. With this understanding we see that when $\nabla$ crosses the integral sign it must be made to change sign and refer only to the part we have called $R$. Thus

Generally speaking $R$ can and will be put as a function of  and for this we adopt the single symbol $u$. Both this symbol and the convention just explained will be constantly required in all the applications which follow.

10. Now let $Q$ be any function of the position of a point. Then the potential of the volume distribution of $Q$, say $q$, is given by:—

the extent of the volume included being supposed given. We may now prove the following two important propositions

The latter is a corollary of the former as is seen from equation (9) Section 4 above.

Equation (19) may be proved thus. If $P$ be any function of the position of a point which is finite but not necessarily continuous for all points

is always finite and if the volume over which the integral extends is indefinitely diminished, so also is the expression now under consideration, and this for the point at which is this remnant of volume. This in itself is an important proposition. The expression, by equation (17), $=-\iiint \nabla uPd\mathfrak{𝔰}$ and both statements are obviously true except for the point $\rho \prime$. For this point we have merely to shew that the part of the volume integral just given contributed by the volume indefinitely near to $\rho \prime$ vanishes with this volume. Divide this near volume up into a series of elementary cones with $\rho \prime$ for vertex. If $r$ is the (small) height and $d\omega$ the solid vertical angle of one of these cones, the part contributed by this cone is approximately where $P_{\rho \prime }$ is the value of $P$ at the point $\rho \prime$. The proposition is now obvious.

Now since

we see that the only part of the volume integral $\iiint uQd\mathfrak{𝔰}$ which need be considered is that given by the volume in the immediate neighbourhood of $\rho \prime$, for at all points except $\rho \prime$, ${\nabla }^{2}u=0$. Consider then our volume and surface integrals only to refer to a small sphere with $\rho \prime$ for centre and so small that no point is included at which $Q$ is discontinuous and therefore $\nabla Q$ infinite. (This last assumes that $Q$ is not discontinuous at $\rho \prime$. In the case when $Q$ is discontinuous at $\rho \prime$ no definite meaning can be attached to the expression ${\nabla }^{2}q$.) We now have

equation (9) Section 4 being applied and the centre of the sphere being not considered as a singular point since the condition of Section 4 is satisfied, viz. that . Now putting $P\text{ above}=\nabla Q$ we see that the first expression, viz. $\nabla \iiint u\nabla Qd\mathfrak{𝔰}$ can be neglected and the second gives

where $\overline{Q}$ is the mean value of $Q$ over the surface of the sphere and therefore in the limit $=Q$. Thus equation (19) has been proved.

When $Q$ has a simple scalar value all the above propositions, and indeed processes, become well-known ones in the theory of gravitational potential.

We do not propose to go further into the theory of Potentials as the work would not have so direct a bearing on what follows as these few considerations.