OF FLUIDS ON THE MOTION OF PENDULUMS. 



[39] 



we get 



Q,-Q.-{2- I + *- , ...+ (20- 1 }f = |l°g(i)-J-Si.* 



The equivalence of the expressions (87) and (89) having been ascertained, in order to find 

 the relations between C, D and C", D", it will be sufficient to write down the two leading terms 

 in (87) and (89), and equate the results. We thus get 



C + Z> log r = ttC" + irD" log r + ZirLl' log (1), 

 whence 



C = 7rC" + 27r log (£).£>", D=nD" (90) 



There remains the more difficult step of finding the relation between D' and C", D". For 

 this purpose let us seek the ultimate value of the second member of equation (89) when r 

 increases indefinitely. In the first place we may observe that if Q, Q' be two imaginary quan- 

 tities having their real parts positive, if the real part of Q be greater than that of Q', and if 

 r be supposed to increase indefinitely, e Qr will ultimately be incomparably greater than e Q r , or 

 even than log r . e Qr , or, to speak more precisely, the modulus of the former expression will 

 ultimately be incomparably greater than the modulus of either of the latter. Hence, in finding 

 the ultimate value of the expression for F 3 (r) in (89), we may replace the limits and ^ir of 

 <o by and an, where to, is a positive quantity as small as we please, which we may suppose to 

 vanish after r has become infinite. We may also, for the same reason, omit the second of the 

 exponentials. Let cos w = 1 — X, so that 



sin' to 



= 2X ( 1_ ^)' dw = 



d\ X dX 



then the limits of X will be and X 15 where \ 1= l - cosw!. Since log 1 1 J ultimately 



-■m\r x 



vanishes, and H 1- ...becomes ultimately 1, we get from (89) 



4 



limit of F 3 (r) = e mr x limit of f X, (C" + D"log2\r) . 



If now we put X=X'r -1 , we shall have and Xi»" for the limits of X', and the second of 

 these becomes infinite with r. Hence 



limit of F 3 (r) = (Zr)-ie mr /"°(C" + ZT log2X') e- mV X'-Mx'. . . (91) 



•'0 



/*CO 



Now / e~"x-ida! = 7r^, and if we differentiate both sides of the equation 



/' W €- ir a? s - 1 d ( »=r(s) 



■ A demonstration by Mr Ellis of the theorem 



L 



2 logsin6de= jjlog(i) 



due to Euler will be found in the 2nd volume of the Cam- 

 bridge Mathematical Journal, p. 282, or in Gregory's Ex- 

 amples, p. 484, 



■f The word limit is here used in the sense in which f(r) 

 may be called the limit of <H r ) when the ratio of <p(r) to/(r) 

 is ultimately a ratio of equality, though /(r) and <p(r) may 

 vanish or become infinite together, in which case the limit of 

 <t> (r), according to the usual sense of the word limit, would be 

 said to be zero or infinity. 



