926 
PROFESSOR A. CAYLEY ON THE SINGLE 
We differentiate logarithmically in regard to u'. Observing that 
K du = 
\/&fdx' 
_____ = h^ dx ' 
y/a-x'.b-x'x-x'.d-x'’ </X! 
suppose, the first equation gives 
A'u A^ur-v/) ^A'(u+u') K^/X' cl Vi 
A u 2 A(». + u ') 2 A(« + v!)' 
and if for a moment 
\/af d; 
, is put = P(« — x')-\-Q(d—x), 
then 
V], = 1, x-j-x , xx 
1, a-\-cl, ad 
l, 6+c, be 
Qf 
(a-a^)Vi’ («— at)Vi 
But writing 1 x —a we have 
Q(d— a), —— Qf= 1, o+rc, ax , = (a — b)(a — c)(d—x), =—be (d—x), 
1, a-\-d, ad 
that is, Qf= — bc(d— x), or 
Hence the equation is 
A_ Ino- V* bcfrZ-aQ 
dx' * a — x! (a — x )y 
and similarly 
A'(¥) 
A(m') 
■ 
v s Q 
I ( 
_L 
A '(u + ri) 
A(u + u) 
_2Kbc 
_ a/ at' 
B'OO 
B'(w— m') 
B '(u + u) 
2Ivca 
B(w') 
B(«~ %') 
B (u + u) 
~ a/ af 
C'oo 
C \u—u') 
C '(u + ii) 
2Kah 
C(tO 
1 C(w— if) 
C (ii + ii) 
~ a/ af A 
D'(?(.—?t') 
D \u + u') 
2K 
55. Multiply each of these equations by du, = « ^/X 
equations such as 
d—x 
L (a-a')Vi’ 
7 d—x 
0 h—x )v 3 ’ 
d—x 
(c-aj')v 3 ’ 
, d—* 
(d — x')(x—x') 
, and integrate. We have 
