580 Mr. H. Bateman on the Velocity of an 



maybe used to express % as a known function of a. Putting 



% = /(«), /?= = we have the integral equation 



M "i-rrz ■•■■ (1 °> 



/ -?L— -a 2 



\x) 



from which to determine v(x). In this equation the value of 

 /(a) is given from a = to oc.=/3; and since % vanishes for 

 the infinitesimal path which is a tangent to the sphere at the 



origin, i. e. when e = 0, «= yy =A w ^ have /(/3) = 0. This 



equation may be simplified by putting 



x 

 v(x) 



it then takes the form 



/■ (a)= «r 4 (log ^ 



This may be identified with a well-known equation solved 

 by Abel by putting 



«2 __ _ ~# _ f 02 _ * 



a — - , t? — , p _ -, 



sea 



We then get 



^(log*)=2«<K0, /W=F(,). 



= ) ^=. • • • • • (11) 



ys — t 



and the condition that F(s) = when s = a. 



Now a necessary and sufficient condition that this equation 

 should have a solution <j>(t) which is continuous in the interval 

 (a<t<b) is th.it F(s) be continuous in the interval (a<t<b), 

 that F(a) = 0, and that 



" ¥(s)ds 



£ 



y/x- 



have a continuous derivative throughout the interval (a< t< h). 

 Under these circumstances the solution of the integral equation 



