26 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
Expanding, we have 
2 = *0 + + 4 { S 2 i 1 '* SL') f ^ 
+ 
| l TO"j4“(/ / ss ' dx/JyXgi) I 
z 0 + 
Our next stage is to evaluate the operation 
s ( r -'d^£f z <>- 
Let us put 
4^1 - *Ts> 
sV 2 = X? — 1, s v 3 = x s (x s 2 — 3), 
and s v P = the y)th function of x s as defined by (xv.). 
Let e s be a symbol such that e/ represents s v p . Then we shall show that 
cl? 
^4 Tss ' dxdxJ z ° ~ 2 oi S, ( r ss ,e s e s , 
We shall prove this by induction. 
By (xii.) 
s^m +1 — “Ts s^m s^m- 1? 
or 
and by (xvi.) 
m +1 - <- m 
— x , e, — m e 
m— 1 
4 5 
cLv„ 
— m or 
dx s 
Now, let x (o) be an y function of e s 
= S(A ? e/), 
if we suppose it can be expanded in powers of e s . 
Then 
de s m 
w- = m e s 
cue. 
i/i—i 
cl 
d x(0 = s( a,1m 
= S(A q qer l ) 
~ S(A 3 (x a e/ — e/ +1 )) 
= £C*S(A ? e/) — e* S(A ? e/) 
= “ o) x(o) • • • 
Similarly 
Now suppose that 
tf 2 
UT x(u » O') ~ (ax —e,)(ay —e y ) x(o, <u) 
^ 1/tX/ gCL'X* 
(lxxiii.). 
(lxxiv.). 
(lxxv.). 
(lxxvi.). 
| Sg^r^/ ( ].. ( £ r ^ | z o — 2 o{^o( ,? ’44'00')] j 
