OF ROTATING LIQUID CYLINDEES. 
89 
Suppose this value substituted for in equation (94), and in the equation so 
olitained equate the coefficients of the various powers of For any power of rj 
greater than the fourth it will be seen that equation (94) may he replaced l)y 
^0 {v — 5^o) - — S oC„ (^o”' + i"') — 0- 
n = — 30 n = 1 
The equation obtained by equating to zero the coefficient of any power of t) greater 
than the fourth will therefore he of the form 
a linear homogeneous function of . . . = 0, 
and there is an equation of this form for every value of n greater tlian 4. 
system of equatious can only he satisfied by taking 
This 
_ P — P — 
oVr, — oV'r, — — . 
(96), 
(97) . 
(98) 
We may therefore assume for (equation (95)) an expansion of the form 
~ • • • 
in which is written for ^C,,. 
F.rom equation (82) we have 
§0^1 (3fo" — Y) = fi (if ioV — 4“^" - W). 
and from equation (87), 
From these last two equations we obtain 
§ 0^1 (34® - ¥) = 4 mr - m - ¥v (4'^ + v^) + (4 + v)- 
With the help of equations (97) and (98) we can write equation (94) in the form 
iv - m m +144) = 3%4® + 4 (w^® - w) 
— V'v (4® + v^) + H~v (4 + v) 
+ -4(4® + V) + ^4 (4^ + v'") + *^4 (4® ■+■ t) + 4 (4 + '>?) + 4- • (99). 
It is clear upon examination of this equation that the equations found upon 
equating the coefficients of rj, rj~^, &c., will contain only terms multiplied ])y 
dg, c/(j, d_j, &c., without constant terms. We therefore take 
4 = 4 = ~ = 0. 
Equation (99) now contains only even powers of rj. Before we can calculate tlie 
coefficients of these powers we must obtain series for 4® ^,4 By squaring- 
equation (93) we get 
2 000 
9rj" 
= 9,* - 20 ,= + + 
Next we have from equation (82) 
4® + T — I (4 — !)• 
Squaring this, and subtracting 24®p® from each side, 
VOL. CO. —A. X 
+ 
(100). 
