18 



C. V. L. Charlier 



+ 00 



(36) b{x) =jdp A(p)<ï)(aj+p) 



— 00 



If, for instance, the character in question is the magnitude of the stars, the 

 equation (35) gives the number of stars of the magnitude x and (36) gives the 

 mean distance of stars having the magnitude x. It is, at least theoretically spoken, 

 possible to determine the left members of (35) and (36) and the mathematical pro- 

 blem is then to determine the unknown functions A and <ï>. 



In practice it is found difficult to determine the mean distances of the stars 

 in such numbers that the form of the function h[x) can be considered as known 

 with a sufficient degree of approximation. It will then be preferable to determine, 

 instead of h[x), the form of the function 4>. Then the corresponding problem is 

 mathematically to determine, from (35), A(p) when a{x) and ^{x) are considered to 

 be known. Both of these problems have been solved by Schwaezschild — A.N. 

 4422 (1910) — with the use of the integral of Fourier. 



l:st Problem. a{x) and <I> being known to determine A(p) from (35). 

 Multiplying (35) by ^ e^""* dx and integrating we get 



1 f -, /X Xini 1 



+ 00 -J- 00 



, dxa{x) e = -j=. \ \ (;te(7pA(p) ^[x\^) e 



— 00 — 00 — 00 



Changing the order of integration we have 



+ 00 + 00 + 00 



1 / , , . xm 1 / 7 A / \ — P^i- I J /HI I \ P'"*' 

 'dxa{x)ie = / dp A(p) e ax(P{x-j-p)e . e'^ 



l/27tj 1/2; 



— 00 — 00 



Observing that the letter integral is independent of p, we thus get, according to (33) 

 (37) a-' (w) := 1/2^ A""' (-co) . 



Hence we have 

 (37*) A-V)" 1 a-H-«) 



and from (34) 



y 2k <e>-'(— 0)) 



+ 00 +00 

 1 r —1 —xMi 1 r a~^((o) .rojt 



— 1/ 



e 



(38) ^{x)=-— di>^^ (w)e =-/cZa) 



— CD — 00 



which formula gives the complete solution of the problem. 



2:d Problem: a{x) and b{x) being known to determine A and <ï>. 



1 XM'i 



Multiplying (36) by e dx and proceeding in like manner as before 



we get 



