OF ROTATING LIQUID CYLINDERS. 95 
taken over the surface, and therefore hv 
I = i[V </.)!.. (124), 
J 0 
where r is given by equation (123). 
Let us assume a solution as far as d’ of the form 
^ yd-) .(125), 
where a, /3, y are functions of (/> only, a being given by 
“"(l ~ A cos 2 (^) = 1 .(126). 
The value of corresponding to solution (12 5) is 
and if we substitute this value for r\ and the similar values for r and 7 -® in equation 
(123), and equate the coefficients of 9 and 9'\ we obtain 
- 0-/3 = 2 (a^ cos 3(/) — - 3 -a cos (/)).(127), 
lyf — |-/3^ = cos 'dff) — |a/3 COS (/> + COS 4^ — -j-ocr cos '2(f) + “5 
whence, by elimination of /3, 
ly “ 8 (a’ cos '2(f) — cos (ft) [a? COS '2(f) — %ct COS (b) 
+ cos 4^ — -y^a' cos 2 f/) + “ 504 ^- 
We can eliminate (f) from this equation by the help of equation (126). The 
resulting value for ^ contains only even powers of a; if we simplify this, and 
transform the numerical coefficients to decimals, we find 
by = 68-la« — 3l9-8a^ + 264-4a3 + 159-4. 
Now we require to find the coefficient of 6" in I (equation (124)), and this is 
|-[ a'^yd(f). We therefore require to know the value of j od"'d(f) for n = 2 , 3, 4, 5. 
This integral can easily be evaluated for all positive integral values of n ; the values 
which we require at present are as follows :— 
, 
a = 2, 3, 
4, 
5, 
— [ d(f) = 4'6 l/'O 
TT Jo 
70-1 
305-3 
Substituting these values, we find at once that ■§ odyd(f) is a positive quantity. 
