510 The Rév. R. Carmichael on Methods 
employing partial transformation as before, we get 
r sin 6 dO =e- *™°dz, 
or 
e 9tan 6 dd= = 
Thus the variables are separated, and the equation is reduced to 
an integrable form. 
6. If the equation to be integrated were 
¥y 
y(yda —axdy) = (v+y)?.e = da, 
we have 
- rsin @dO= (sin 6+ cos 8)%e-™? da, 
or 
emmtan Od _ dx 
l+sn2e 2? 
where, as before, the variables are separated, and the equation 
reduced to an integrable form. 
7. There are some cases in which the solution of partial dif- 
ferential equations may be facilitated by transformation to polar 
coordinates. In these cases the partial differential equations 
reduce themselves to complete differential equations im a single 
variable. Thus the ae 
ris (0) thea 4 
is reduced rs once to the form 
OF =Antan"O+ Ann, tan*-! 6+ &e., 
which is easily integrated, and 9 replaced by z and y. 
8. Let it be proposed to integrate the system of simultaneous 
partial differential equations: 
du 
= =cy +b, | 
daz 
we | 
a =a,2+ 0,2, 
du 
Ea bv + ayy. 
Multiply the first equation by 2, the second by y, and the third 
by z; add; and we obtain 
(22 + ts 2£) u=2(ayz +b,2@ + c,xy) 
ae Yayt OME NOPE TMEY); 
