112 Proceedings of Royal Society of Edinburgh. [sess. 
Hence the force <£ co' becomes mcf>f~ 1 (D = 7 n cf)f~ 1 (D + mVef _1 o>. Thus 
we see that the force on an elementary area, which before strain is 
a), is a linear vector function of w, but even in the case when there 
is no molecular couple, this function is not in general (large strain) 
self-conjugate. If there be no rotation, this force 
= m Tffip ~ ho + mV e\J/ ~ 1 w 
= ^dwto + Y 6 m + . 
If now the rotation take place this force merely rotates with the 
body, and we get for the force rw on the area, due to the^ strain 
iH )t\ 
TO) = q(^dW(o + Y6(t) + mY€xl/~ 1 (D)q~ 1 . . . . (38). 
It is not hard to see from the above that the couple per unit 
volume of the unstrained body is 2mqeq~ 1 . The force per unit 
volume of the unstrained body can be shown by means of equation 
(3) to be r A. From these we can write down equations of motion 
in which i}/ occurs. To obtain equations of motion in which only 
p and its derivatives occur explicitly we must adopt a different 
method. 
Consider w a given function, not of i]r as above, but of «// 2 or fx 
(Tait’s Quaternions, 3rd ed., § 381). Let 
^O) = l/^ 2 0D = = V j^So) V 2 SpiP2 .... (39). 
In assuming that w is given as an explicit function of the 
coordinates of \I>, we are following Thomson and Tait, for these 
coordinates will be found to be the A,B,C ,a,b,c of Appendix C of 
their Nat. Phil. 
To see the enormous advantage of employing quaternions in such 
questions as the present, it is only necessary to compare the pro- 
cesses and results of the present investigation with theirs. The 
results below are considerably more general than theirs, and yet how 
much less cumbrous. 
By the methods already so often applied, it is easy to prove 
both the following identities : — 
S8 x^x' _1 ^ = s x' s x^x"%''^ 
= sSx'xCx' W’f- 
