122 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Again k 2 is negligible and 



rp f T ,15 



T=--K\0g- K --K. 



If we had desired merely the total time T -f- T' = t — 2k for the 

 swing from rest to point d'arret, we could have obtained the result 

 more directly by the theory of residues. The integral 



du 



taken around the path in Figure 2 which passes through infinity, is 

 equivalent to the sum of two integrals 



du „ du. 



U AIL 4K. U 



~ du du, 



I 2 u i u I 4k+ / 2 u u I 4 K ' 



where the first is taken along the direct path on the upper side of the 

 junction line and where the second is taken around that pole of the 

 integrand which lies in the upper half of the upper sheet of the Rie- 

 mann surface. As the calculation of the position of this pole is neces- 

 sary for the work of the next section we shall now check the above 

 value of T + T'. 



The denominator of the integrand vanishes when 



. u 3 . u 2 u 2 A , 4/A 



This gives one zero at the branch point u = 0; the integral to u = 0, 

 however, is finite as the root is simple and the branch point has two 

 sheets. The other roots are determined by the cubic 



— KW 3 + U 2 + 1 = = — K.(U — ~ K — K)(u 2 + KU + 1). 



To terms in k the approximation gives the three roots 



1 , . K 



U= - + K, U = ± I — -. 



K 2 



The first lies in the under sheet and does not concern us. The root 

 u = i — \k lies in the upper half of the upper sheet, and the residue 

 of the corresponding pole has to be found. The expansion 



u , u ( 2k 2k 2 

 2k 2k \ u w 



