518 
EXTRACT FROM A LETTER TO MR. MERRIFIELD. 
[538 
that is, 
*-/?• /-/?. 
or, writing a = (f)h, 1 
7 = yli, we have 
p = x(f>h + yh + yji, 
where 
? f dh f yh . c 
q=hx+y !Wh + i^h 
X(f)'h + y + x'h — 0. 
The last equation gives h as a function of (x, y), and the values of p, q 
such that dz = pdx + qdy is a complete differential, so that we obtain z 
integration of that equation. 
A simple example is 
that is, 
whence 
p = \h-x — hq, q = — lix + y log h, hx — y = 0, 
P = -h y ~> ( t = -y + y\ogl> 
we have 
or 
as it should be. 
z = hy-^l~i y\ 
r = Æ> «=*--, t=\og V -, 
R-L 1 * 1 
X 3 ’ x 3 > ? x’ V ’ 
- (--£ 
/3 y 8 V x, 
are then 
by the 
Cambridge, 28 July, 1871.