645 



.When moreover we put herein c = — , in which Z) is a new 



constant, we get exactly (lo)^). 



II. From (11") we can deduce a differential equation for f{a,t). 

 When viz. we expand in the first member f [a- — x,t^) in a series of 

 powers of ^,, we find : 





-f . . . = f{a,t,) + t^ 



dt. 



1) We may give this proof in a slightly different way yet, viz. by using only 

 (lla). Take in (Ila) ti = t^ = . . . = tn = t, and make n great. As is known from 

 the quoted proof of Bessel (1. c.) : 



in which 



X 



f{c(,t) = ^ j \yi{k)\"ro,a).dt, 



00 



00 



X (A) =^ I ƒ (.^'jT) cos xk dx . 



By expanding cos x>^, we get: 



X 



t{k)=l-^^,jx-f{x,x)dx^... 



When T is small, the coefficients of the powers of a diminish rapidly and are 

 small, so : 



00 



X a) = 1 - ~Jx^f{^v.r)dx = 1 -p,X' 



— a 



and : 



while P2 is small and n great. Finally : 



X a" 



a,t) z= — cos a X «-/'^"'^ = e '*^'" 



2:n:J 2l/ jp.n 



/( 



Now riT = t. Further take — = D, then we find again the formula (la). Herein 



T 



D is independent of t, while from the found form of f follows that p^ is pro- 

 portional to T. 



This proof shows that always the same f {a,t) is found, independent of the 

 choice of the "elementary'' f in the first member of (Ila). The value of D however 

 in this case depends on the form of this function. 



47 



Proceedings Royal Acad. Amsterdam, Vol. XX. 



