of Edinburgh, Session 1877-78. 529 
(4.) But we have, by differentiation, from the second equations 
of § (3), 
dxdy 
d Al . d JL=A\.iVv 
dx dy dy 
1 
*£?q + *LL .<k-±V JV v 
dydx dy dx dx 
Subtracting, and noticing that 
dq 1 dq = _ {dq dq 
dx' dy ^ dx ^ dy ’ 
we have 
SV. q-^f ld i = Y.(j d - i4-\vv = V. V IkV) Vv , 
1 dx * dy \ J dx dy) x ; 
or 
2S(&Vv) . Vv = Y.V (&V) Vv . 
Three like this give at once 
(Vvf= -V 2 v 
or 
0 = 2uV 2 u-(Vu) 2 = 4u%V 2 (u h ) . . . (O (21).) 
(5.) But if, instead of combining the last set of three we equate 
to zero the scalar coefficients of i, j, k separately in each, we have 
three equations of each of the following forms : — 
2 
dv dv _ d 2 v d 2 v 
dx dy dxdy * dx 2 
+ 
d 2 v 
dy 2 
Transformed to u , they become 
The integrals of the first three are obviously 
\ du _ r , 1 du _ v , 1 du 
u 2 dx ’ u 2 dy 5 u 2 dz 
where the right hand members are functions of x, y , z respectively. 
Thus 
- = - X-Y-Z 
a 
