388 
Proceedings of the Royal Society 
To the foregoing Professor Cayley has kindly made the follow- 
ing additions : — 
The investigation may he carried further : writing for shortness 
u 3 > u ±> & c -> in place of ^(3), ' V P(4), &c., the equations are 
— 1 
u . 4 — 2% , 
% — 3 % + 6 % + 1 , 
u Q = ^ u b + + 1 2 % 
% = 5 % + 10 % + 1 5 % + 18 %+ 1 , 
and hence assuming 
U = % + UjX + u b x 2 + u 5 x 3 + u 7 x* . 
we have 
u = YZ^ +u 3( 2x + 6x2 + 12x3 + 18* 4 + 
+ %(3a; 2 + 8a; 3 + 15a? 4 + 22a; 5 + 
+ %(4a; 3 + 10a; 4 + 1 8a? 5 + 26a; 6 + 
+ %( 5x 4 + 12a? 5 + 21a; 6 + 30% + 
and hence forming the equation 
u 
a?" 
(f- x) 2 = X 2 + ^ + ^ + ^ xb + 
+ %(2a? 3 + 4a: 4 + 6a; 5 + 8a: 6 + 
+ %( 3a; 4 + 6a; 4 + 9a? 6 + 1 2a; 7 + 
• ••) 
...) 
...); 
du 
whore u denotes ^ , we have 
x 
u-u ( T -T)2 ~ l ~ x 2 ^ 3 + U * X U[>x 2 ‘ ‘ )(^ a? + + ^ + 1 8a? 4 . . ) 
1 
j-— ^ + w(2a; + 6a; 2 + 12a; 3 + 18a; 4 + . . . ) ; 
or, what is the same thing, 
1 
x A 
1 f 2a? 2a:* ) 
f~i-x 2+u \ (i-*) 3_ a -*) 3 ( \+ x ) / ’ 
2x* 
U-U 
(l-x 
