308 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
where 
S = 
s (2s - l)(2s - 2) (2s - 3) 
2.3.4 
and T stands for the remaining terms, involving 
t, u, v, . . . 
y, I , 6’, t, . . . 
Considering 
to be of the order 
'25 4 ’ 83 
we may reject the term which is of zero order, and write 
1, i- 4 i 
P = T 
2. r 3 . 
3 ' 5 
+ r a_i +S -T. 
Hence, rejecting terms of order less than f (which have, however, to be retained in 
obtaining the subsequent approximations), 
(p - ?) L - 
(q - rf 
Aa.3 . 
3 ' ’ 
t.e., 
L - r + 2 r; 
= (2 qr - fr 3 ) ; 
(p -q) — (q- r ) 2 = 
- + r 3 - 2 + S - T 
o 
o 
— 
when q is infinite. 
Again, writing for S its expanded value, viz., 
s 1 
Q 
o 
I 1-1 o2 ~ 
* n l 2 * , j 
we have 
(p-q) j f 2 qr 
(q - rf II + 2 q'r - f q + & 
|r 3 - £g* 
Order f 
3 . 
2 > 
-i, 
q ^' 2 — - — 5 s + H* 3 — £$ — T 
i; 
< i, 
rejecting the terms q~h' 3 , q~h A , ... in the expansion of (q — rf because the order 
of none of them is superior to zero. 
We now write 
q = (r - 0 X s/rf , 
so that 
2 qr — f-r 3 — ^cp = (2r 3 — -f- 2dpr 2 ) — f r 8 — (fr 3 ~ Adp 4 + 4 Opr* — 
= - 2 Opr- + 1 6*r\ 
H ence 
(p - q) 
(q - r T 
- i(q ~ rf 
— (q — r) z !> = -<; 
— 2 0*r* + 2qb' ~ |q + |-s 4 ' 
Order 1 ; 
3 . . 
4 5 
/ 3. 
^ 4* 
