1890 - 91 .] Mr A. M'Aulay on Quaternion Differentiation. 107 
position, p the coordinate in the strained position, and p' + Sp' in 
the additionally strained position. Let V have the usual meaning 
with regard to p and v', the same meaning with regard to p (so 
that in the notation of p. 103 above, V' = p, V). 
Let at a given point co be the vector area of an elementary inter- 
face in its strained position. Then (Tait’s Quaternions , 3rd ed., 
§ 507) the force due to stress on this area will be <£a>, where is a 
linear vector function not necessarily self-conjugate. is a function 
of p', or of p only, and does not in any way depend on <o. The force 
and the couple per unit volume of the strained body due to this 
stress are respectively (Mess, of Math., vol. xiv. p. 29)^^! (or 
<£A', as it might in accordance with the meaning of A, explained on 
p. 104 above, be written) and (or YV \<f>p^ or YV 1 ^>p 1 as it 
might be indifferently written.) Let now Viv',ds' stand for the 
unit normal and element of surface at a point of the boundary of the 
portion of the body considered, in its strained state. Let also df 
stand for the strained volume of the element d$. Then noting 
that by Tait’s Quaternions , 3rd ed., § 384, the rotation of the ele- 
ment due to the additional displacement is YV'Sp 72 , we see that 
the last equation gives 
fff Swds = - ff&8p <f>\Jv'ds + ffids + fffSeffSp'ds, 
where e is put for Y£<££/2, and .*. (equation (5) above). 
<£ = ?+V e ( ) (15), 
where </> stands for the pure part of c£. 
Transforming now the surf. int. into a vol. int. by equation (3) 
above, we have 
f/S8p'<f>Uv'ds' = fffSSpffk'dd 
= ./Z7X s fy>i<£vi + S Sp'faVJds. 
Combining this with the other volume integrals, and noting that 
- S3pi<^vi + SeviVi = - SSp^v/, 
we get 
fff8ivd<s = . . . . (16). 
There is an important result that flows at once from this equation, 
true whether the strain be small or great, and which, I believe, has 
not hitherto been noticed. Into the expression just obtained for 
