18 
MR. E. T. WHITTAKER OR THE CORNEXIOR OF 
Then the differential equation becomes 
Hb ^ ^ 2e, .2" ' + “ + cZ 22" * + ... + <_i, 
where c^i, do, . . . d,,,_i, are new undetermined constants replacing the c’s. 
This can be written 
^ — S (n + 1) dz/dw + “ + . . . + Ln-i, 
1 ^ .. n — 2 1 
= = 8 (n + 1) ^5^^* + + ■ • • + 
tr = {z — ei) {z — Co) . . . {z — e,,+o), 
or 
(i )7 
where 
arid where k^, ko, . . . k,i_i are new undetermined constants, replacing di, c^,. . . <:4-i- 
Ifzh .as its injiyiity at a double point of one of the substitutions, we get a slightly 
different form of the equation. 
In this case, one of the e’s is infinite. Let 6,^+2 = o®- Then, near z = oo, the 
expansions are of the form 
z = 
{t - h) 
2 T* . . . and II (z) — -f- . . . , 
whence, by the same reasoning as before, M^e find that 
/l+ 1 
i {t, z} = A 2 
Put 
L_ 3 — wz” ^ ^ + • • • + C»-1 
,.= 1 (z - e,.f ^ ® (z - ej (z — e.) • ■ • (2 — e„+,) 
dz 
IV 
=f 
v/(z - ek) (z - e.) ... (2 - ««+i) 
Then the equation becomes 
{t, iv] = —^ + dd" " + da^^ ^ + f-u 
where again the quantities c/j, dg? • . • <^,i-i> are undetermined constants. 
This can be written 
i {b = o ' /A + .. • + 7 
or, 
where 
8 (?i + 1) dzjdw 
n — 2 1 dhi 
- fi- kiZ'^ ^ + ^2^^” * + ...+ I'n-i . . . (2), 
8 {n + 1) u dw 
xd = (z — Cl) (z — e,)... (2 — e„+i). 
