OF ROTATING LIQUID CYLINDERS. 
77 
We are in search of a solution for ^ expanded in powers of c in the form 
i = 'Q + + ^^3^^ +. 
in winch . . . are functions of r]. 
Substitute this assumed solution in equation (36), and we obtain 
V (wq + + WgC^ + . . .) 
= a~ + c { T]’‘ + Vq‘ + u^c + \n (a — 1 ) c" 
+ nuQ‘~^ UoC- + {n — l) {n — 2 ) 
n{n — I) u.,c^ + c^} 
d- terms of degree 4 and higher in c. 
Equating the coefficients of the various powers of c, we obtain 
7]Uq = a- 
V^h = V’‘ + 
-- n’i Ti 1 
'qll^ = nil 
n 
n 
[n — 1) ?q”'~ iq' + ^ 
-qu^ = Y?i [n — I) {n — 2) zq" n {n — 1) Uq‘ ^ ;q, kc. 
Solving these equations in succession, we obtain 
a~ . «■“ 
V 
«i = >)” ‘ + 
-jq = n«'' + 
V 
?i +1 / ’ 
^,3 = in (n - _j_ 
9(. (3a — 1) fd"' 
77 
*)^ort + l 
iq = (n — 1 ) (?i — 2 ) a~“ ® 17 '" ^ + 
2.-1 , 
-V 
a 
4«-0 
+ terms of lower degree in 77 , &c. 
Hence we obtain at once 
V, = C + 77 P j - c + 4 (« - 1 ) «■=-* (f' + >,") c= 
-f (n — 1 ) (n — 2 ) 
■J 
=: C + Trpj — r~ + (?^ — 1) a~"' r’’- cos 
+ ^ (n — 1 ) (n — 2 ) rt~'‘ ^ (Ayi'’- cos '2n6 + 
A = 77 {a* + na-”- ~ c- + f ;i (71 — 1) -f • 
(38), 
and, except in the sj^ecial case of ?i = 1 , a = 0 . 
In this way wm can, Avhen c is small, wnite down the potential to any required 
degree of accuracy. 
Deformed Elliptic Cylinder. 
§ 13. As a final Illustration, we shall find the potential produced by a small 
deformation of the surface of the elliptic cylinder 
= .(39). 
Let the deformed surface be 
