120 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



positive. Hence the positive sign must be taken with the radical. 

 When x = 0, u has become positively infinite.. The time of fall to 

 the origin is therefore 



2k 2k 



vu 2 + 4ku 



Introduce the substitution 



r = 



u 



m + 4/c' 



u = 



4k?/ 2 



(13) 



Then 



J" 



Jo 



4k efy 



8kV+(1-2/ 2 )(1-?/)- 



The integration may be performed by partial fractions. It is 

 necessary to factor the denominator approximately. One root is 

 nearly equal to —1, the other two are imaginary. The real root may 

 be developed in powers of k 2 . The first approximation gives 



y = - 1 + 2k 2 . 



The denominator is factorable as 



(1 + 8k 2 ) (y + 1 - 2k 2 ) \jf - 2(1 - 5k 2 )*/ + 1 - 6k 2 ]. 

 The reduction of the integrand to partial fractions gives 



1 if - (1 -ok 2 ) - (2 -7k 2 ) 



k(1 





y + i 



IK 



For the integration 



= K (1 - 2K 2 ) 



i]0g f 



if - 2 (1 - ok 2 ) y + 1 - 6k 2 

 (y + 1 - 2k 2 ) 2 



2(1- 5k 2 ) y + 1 - 6k 2 



2 - 7k 2 ~ 1 y - 1 + 5k 2 " 

 H — - tan 



2k 



2k 



Substituting the limits and 1, we have 



r = k (1 - 2k 2 



log 



- k 2 1 - 3k 2 \ _ 2 -- 7k 2 



1 - 2k'V 



2k 



/ - 1 5k - x 1 - 5k 2 



(tan g-+ tan -^- 



The k 2 is now everywhere negligible, and 



7T 



-^l + l- 



\ 



