OF FLUIDS ON THE MOTION OF PENDULUMS. 



[43] 



Let 



m 



m 



1 I s . 2 2 . 8 



2 



+ ... = ML, 



in 



8 



m 



5L 



1* l 8 .2 a .3 



»4 



J+... -if,', 



tn 



m <? _ m .«? + = TV 



tn* 



1 2 .2 2 1 2 .2 2 .3 2 .4 8 



tn 



+ ... =M e , s 



+ ... = .My, 



Mft* 



iTa* " i 2 . 2"! s*. * s * + '" = N " 



nr 



tn 



F' S ' 1 ~I^i^' S ' 3+ •" = Ar °'' 



tn' 



SU- 



m' 



,Si + ... -2V/, 



(103) 



1 2 .2 9 1 2 .2 2 .3 2 .4 2 



log, nt + A = i: (104) 



then substituting in (101), changing the sign of \/ - 1, and arranging the terms, we get 



&+\/^A'=i+— 1 i - - (105) 



m - ^ jf;+ z (1 - mj) + n;+{- lm ' - - (1 - iio + jv '} v/ri 

 4 4 



33. Before going on with the calculation, it will be requisite to know the numerical 

 value of the transcendental quantity A. Now 



T- J r'(£) = (r^)-T'(i) = ^i°g.roo 



ds 



and the value of — log T(l + s) may be got at once from Legendre's table of the common 



ds 



logarithms of T(l +«), in which the interval of 5 is .001. Putting l s for the tabular number 

 corresponding to s, we have 



lo g r(l + «) - 1000 log, 10 {Al s - £A 2 /,+ ^ A 3 /, - i A% + ...}. 



ds 



For a = 1 



AZ S = + 16050324, A 2 Z,= + 405620, A 3 4= - 359, A 4 Z S = + 6* 

 the last figure being in each case in the 12th place of decimals. We thus get 



tt ^T'(l) =- 1.9635102, A = + .5772158 (106) 



34. When ttt is large, it will be more convenient to employ series according to 

 descending powers of a. Observing that the general term of F 3 (a) as given by (88), in 

 which Z>'= 0, is 



klj- [l.3...(2i- 1)1 2 



v ' 2.4...2i(4ma)'a4 



* These numbers are copied from De Morgan's Differential and Integral Calculus, p. 588. 



30—2 



