OF ROTATING LIQUID CYLINDERS. 101 
We calculate in succession the value of the radius vector when (f) — 180°, 150°, 120°, 
90°, 60°, 45°, &c. The value of r is a root of (r) = 0, where 
4? (r) = — r” + 1’162 — ‘2llrcos(f) + &c. 
Our method will be to find a value of r for which cR (v) is small, and calculate fi>(r) 
and d^ldr at this point. The true root Ti is then given by 
R=, 
{i')ldr 
When cf) = TT, cfi = 1-162 -f -21 Ir — -LOZm — -0537’^ + -OH/-^ — '0045r5 + -0010-e 
— -00043?’^ H- [-0002;-® — -0001 + ...]. 
When r = 1-8, <h := 1-162 + -380 — 1-319 — -310 + -147 - -085 + *038 — -026 
+ [-020 — -018 — -003 + [-01], 
d^jclr '21 — 1-4 — -5 -j- -3 — -2 '1 — . . . — — 14:, 
therefore K = 1-8 — -002 + [-007], or, II = 1-805. 
When (f) = 577/6, ffi = 1-162 + -1827- — -704?- ~ -OOZ?-'^' -f- •0039;-" — -0011?'® 
+ -00036r' - [-00017-8 + -0000?*^'^ +...]• 
Whenr = ^2 = 1-414, 4>=ri62 + -257-l-408--028+-022 —-OOO + 'OOI - [-001] 
= -000 - [-001], 
therefore 11=1-414, 
When (fi = 77 / 2 , $ = 1-162 — 1-5937- fi- -0147-^ — -OOll/-*^ + [-0002r®] + . . . 
When 7’^ = ’74, r = -859 ; 
fh = 1-162 — 1-179 + '007 - -0003 + [-0000] + ... = — -010 ; 
d^ldr = — 2-5 ; therefore R = -859 — -004 = -855. 
When (f> = 7|-°, fL = 1-162 - -2097- - -4277-'^ + -0467-® + •0127-'^ +-0036?-^ + 'OOOS/’S 
+ -00037-' + [-00017-® + -00004/-'-^ -f . . .]- 
When 7-" = 3, 7 -= 1-732, fl> = 1-162 —'348 — 1-281 +'241 -|--108 +-056 + '021 
+ -014 + [-008 + -005 + ...]=- -027 + ['01]. 
d^P/dr = - -209 - 1 -44 + '42 + '23 + '17 + '08 + ... = - '75 ; therefore 
R = 1-732 — -036 + [-01] = UZl. 
When (^ = 4°, (p — 1-162 — '21 \r — -llO?-' + '0527-® + -014?-^ + -004/-^ + 'OOlr'’ 
+ -0004?-^ + [-000177-® + -00008?--’ +...]. 
When 7 - =2, = ri62 — '422 — 1-640 + '416 + '224 + -128 + '064 + '051 
+ [-05 + -04 + ...]=- -02 + [-1 ?]. 
d^/drz= --211 - 1-640 +-624 +-448 +-320 +-192 +'-178 + [-18 +-18 +.. .] 
= -oi+[?]. 
When r = 1-8, 4) = 1-162 - -390 - 1-328 + -297 + '147 + -076 + -034 + '023 
+ [-017 + -014 + ...]= -02 + [-05]. 
d^P/dr = — -211 — 1-47 + '49 + '33 + '21 + 'll + -09 + ['08 + '07 + . . .] 
= - -45 + [-2]. 
