1890-91.] Mr A. M‘Aulay on Quaternion Differentiation . 113 
Thus we see from equation (30) that 
- i s (x' s x + s x'x)£x~ !< £x 
= -hsmx'tx'-'t- 
Eut we also have by equation (13) above 
8«;= -SS^a«o£. 
Hence from p. 104 above we see that 
%aw = m x -'4>x-\ 
or 
<F= ^x*a*x'> ( 40 )- 
(It is easy from here to go back and prove all our previous results 
over again. Perhaps this, in fact, would be the shortest method, 
but it would not be the most natural. To do so it is only necessary 
to notice that since SSi/^d£ = SS4>£^a£, it can be proved that 
^<3 = ^Gi]/ + ij/^a where the differentiations on the right are not to 
act upon if/.) 
Substitute from equation (27) for V ' in the equations (17) and 
(20) of equilibrium. Thus 
8 - " V , + V , = 0 . 
Noticing that since by equation (11) 
2m\~ 1 w= - Yp'jp'gSoo V i V 2 > 
we have 
my" 1 A = 0 , 
this last equation can be written 
g -|V^x'‘ ] A+m^x'" 1 A = 0, 
Substituting now for <f» from equation (40) we have 
g-iy^X , ~ 1 A + 2 X ^A = 0 (41), 
or 
S+i( 
VWVp 1 
1 P 2 
SV 3 V 4 V 6 S P »' 5 
)s A V 1 V 2 - 2 p‘S V !»aw A = 0 (42). 
i/ 
VOL. XVIII. 
29/4/91 
