644 



In connection with the physical meaning of ƒ (a-, we know that 

 the function / is even in ,/-, and for a small t differs only from 

 zero for small values of x. From the proof of the law of ei-rors as 

 given by Bessel '), we know that under these conditions a product 

 that is built as the first member of (II''), becomes for great values of n: 



f{a,t)da = qe-^^'' da (m) 



in which ff and if? are functions of t still unknown. To determine 

 these, we substitute (III) in (II''). This gives after integration 



From this follows 



'' ''^ ^ '^^ = ^Rg +lKg ^"' ^ '^ ^ ''^ ^ 7wïïTv(ö 



c 

 The solution of the first equation is : tp (/j = - *). 



This, substituted in the equation for (f, gives: 



7 (^ + h) = 



This functional equation is solved by 



When we substitute this in (III) and consider moreover that 



r f{a,t)da = \, 



— oc 



then it is proved that c' = and we find 



0''C 



j{a,t)^\/^^e 



i) Astr. Nachr. 15 (1838) p 369 = Ges. Abb. 2. p. 372. 

 2) Put: 



_ Mt) k{t,-VQ __ ^(^) + M^.) 



''' ^^^ ~ T • '° "VH. " hk{t,)^t,k{Q ' 

 When we put herein successively: ^. = ^i, 2<i, 3ii,...,tben appears: kih) = U^h) 

 so kit) is a constant 



c 

 5r 

 c' = constant. 



3) Put viz. <p(« 1/^=1/ ^ L (0- than Ut^ + ^g) = L{t,)L{L) or L(<) = e^'' in which 



