in the Integral Caleulus. =~ 511 
whence at once . spe 
us (ayz + dyzn+ ¢,2y) + Up, 
where u denotes an arbitrary homogeneous function in 2, Y, 2 
of the order zero. 
It is to be observed, however, that wu» cannot contain any inverse 
powers of x, y, z. For instance, uy cannot be of the form 
aut + By 
yu + oy’ 
as in that case u would not satisfy the separate partial differential 
equations. 
Thus the ultimate form of the required solution is 
u=2(ayetb2a+ cay) +k, 
where & is an arbitrary constant. 
9. Again, let it be proposed to integrate the system of partial 
differential equations 
d?w 
a =™ 
d*w 
dx dy»? 
d*w wy 
dy? 2 
where ), 4, vare given constants ; or, more generally, the system 
dw 24 dy? 
eae al dle 
d*w 
eq at inet 
2 
a =cy? + bx? +. 
Multiply the first equation by a, the second by 2zy, and the 
third by y?; add; and putting the symbol 
"an t¥ dy‘ "dz Vv? 
we obtain 
V(V—1) . w= (aa + 4ba*y? + cy) + (ra® + Quay + vy?), 
whence at once 
w= 7p (aa + Aday? + oy") +3 (Ma? + Qpay +942) +2 +Uo 
where u, v are arbitrary homogeneous functions in 2, y, 2 of 
the degrees unity and zero respectively. 
