OF ROTATING LIQUID CYLINDERS. 
91 
(1 — 
''G - 7 5 
_ 5 50 0 _ 
'•' O — 6 3 3°2 ’ 
We therefore find as the value of 
4375 
i — 1 8 0 „3 _ 2 8 2 5-- 
&2 — 7 V 28 V 
'•*'3 — 5 6 2 5 ^2 
,7 _ 43 . 75 . £ 915 ^ 
ct — SI j. S I ^’2 * 
+ • . . + §3 ^ ih + + 
( 101 ). 
§23. W e now proceed to the determination of ^ 3 . Equation (85) takes the form 
(§0^3 + § 2^1 + fSs^o) iv — Uo) 
5'^0i=l \'^0b3 m 5'-'2i=07 
+ (Sol^,)-^ 3^0 + (So4 + fS2fo) (3^0^ - ¥) 
+ § 0^1 - -IF^o) + SoS 2 lf^o^i 
+ 3^0 + “ 3^'« (^ 0 ” + '>?”) • • ■ • • • 
( 102 ). 
All the terms on the right-hand side, except those in the last line, are of odd degree 
in 7] and of degree 5 at most. The same is true of the terms on the left-hand whicli 
are multiplied by So. The terms multiplied by 83 are of even degree, two at most. 
It is therefore clear that we may at once take 
jCg — 3 C.J — 3 C 5 — 3 C 9 — ... — 0 , 
83 — 0 ; 30 ^ — 3 G 2 — 3 C 1 J — ... — 0 , 
and assume for an expansion of the form 
4 = 56577^- -f 3^3772 + _ -I _ . . . . 
(103). 
AV e now calculate the various series which occur in ecpiation ( 102 ). We have 
5 _2^ 500 
6r) 2*1 rf’ 2437?* 
So {v — iQ 4 = 8^ — “ 9-^3 
3’ 
and from equation (97), 
S2 - 5 fo) = S3 [f (^0^ _p ^3^ _ 16 _p 
= -80 
64 
37577 
This last bracket can be at once calculated from the series of the last page ; we have- 
S 2 (^ — 3 ^ 0 ) ^1 = S 2 (fr^s _ 677 -f 0. 77 “^ + . . .) .... (104). 
Next 
fSo^^ (So4 + fS2Go) + (S 0 G 2 + fS24) (3^0= - Y) = {U2 + |S2^o) (i^i + 3 Go^ 
From equations (93) and (90) we have 
'-2r - A + 27 
10 ' 
3 y 
‘‘V 
lil 
3 
= 9 2 . Y5 , 
”^ 977 ^ 8 V 
( 105 ). 
From equations ( 101 ) and (^89) we liave 
N 2 
