1890 - 91 .] Mr A. M'Aulay on Quaternion Differentiation. 121 
and .-. 8(S3)6*) = (2S3)Se - S2>@S VS V + i-S@SK @)rf s . 
Also 
JJJ'8vS)ds + fffhv.crds 
= - ff/8v S V 2)c?s + ffhvSUv'&ds [equations (73)] 
= ff/S ( $)'V§vd<s [equation (3)] . 
Hence 
aw = fffds | S3) ( V Sv + 8®) - |S3)gS V 8, + J-Sg8K@ | 
But 
V8v + 8® = -8V.v= - VjS&hVv 
Hence 
8W = -Jff S8 Vl (J V V x 3))<fs + jJjT’SgSKgefe 
The increment SK is conveniently considered as consisting of two 
parts — first, 8K„ due to the rotation of the body ; and second, SK„, 
due to the change of shape, i.e., the pure strain. If the rotation 
vector due to 8rj he e, i.e., if any vector <o become to + Yew, the result 
of operating onw + Ycw byK + 8K r must be the same as first operat- 
ing on (o by K and then rotating. In symbols 
whence 
Thus 
(K + SK r )(<o + Yeo>) = Ka> + V € Ko> , 
8K r (o = YcKa)-KY 6 a). 
S@8K r ® - S®eK® - S(£KY € (£ = 87rSe£>(£ , 
whence giving € its value, JY V Srj, 
~ Sg8K r g = JS V SjjVSe . 
07 T 
If we put w for the pot. en. per unit volume we have 
w= -iS2>e= -ls@Kg .... (78). 
07 r 
K is a function of the independent variables e and i f/, where if/ is 
the pure strain given by 
if/ W = — Vj+ V jScjO^j). 
