OF ROTATING LIQUID CYLINDERS. 
87 
Now o)' Avill be a function of 0, and the relation between co^ and 0 is at our disposal, 
this relation being virtually the definition of 6. We have, however, already assumed 
that at the point of bifurcation 0 = 0 and dco^/dO = 0. We thei’efore take as the 
relation between oj^ and 6, 
L — (o^j'Iirp = 8q + S.^0~ -j- fi-. 
ill which Sg = f, and So, 8g, ... are as yet undetermined. 
Tlie value of ^ given by equation (71) is now 
^0 “t" ^§0^1 + (^0^:: + (^0^3 + ^i^l + f^3^o) + ' • • • 
If we substitute this value in equation (70), of which we are supposing it to be 
a solution, we obtain 
V do + ^^u^i + + ■ • •) 
= + f It + d" + (^^0^1 + (^0^3 + l^3^o) + • • •) 
+ (dSg^i + + fsdo) + • • on 
+ 0^ (NO + d" + ^ 0 ) ■ • -(SI). 
.9=1 n = l 
and if we equate tlie coefficients of .successive powers of d in this, we obtain 
^0^ = «~ + f (bo^ + T?0. 
^0^1 d ~ 5^o) = id + ^ (^o" + "^'O. 
1 
+ I h^o) — 5 ^ 0 ) = I (Sofi)" + S n NO 
1 
+ 2^0 N S (^q”' + r)“) 
I 
(§ 0^3 -b iv — 5 4 ) = INbi (80^2 + |8du) 
+ (80^1)^^ (''■ ~ 1) N-i + (80^3 + 58 do) ^ ik'.do'''”^ 
1 1 
+ ^0^1 ^ aC, "1 3^ "1 S (^o" “1 v ')- 
(82), 
(83), 
(S3). 
(85). 
Equation (82) is, as it ought to be, identical with (72). Equation (83) enalfies us 
to determine the constants which occur in fj. These having been found, equation (84) 
enables us to detei'iniiie '’O on in succession. 
§ 21. As far as first powers of 0, we know that the configuration is of the form 
given by equation (6.9). We therefore assume at once for N fh© form 
N = Scgd + cq 
^‘=1 
o 
77 “ 
( 86 ), 
in which c„ is temporarily written for ^C„. 
Since 8g = |-, and since the c’s higher then cq must vanish, Ave find that equation (83) 
takes the form 
Hn — ifo) = 03 (fo®’ + ’)’) + “1 (fo + >)) 
( 87 ). 
