Studies in Stellar Statistics 17 

 A fuucliou F[(.<i) may be defined through the following series 



+ 00 



(30) = a 5]/(ma) e'"*«"^ [i = l/^l). 



m — — 00 



It is supposed that this series is convergent or more simply that the function 

 f{x) is so defined tliat the integral 



+ 00 



(31) l{^)=jdxAx)ë"-'' 



— 00 



is convergent. We evidently have 



lim F(a)) = /((«). 



Multiplying (30) by g-'''^-"»* and integrating between the limits — Jr/a and -|- 7r/a 

 all the terms to the right, excepting one, vanish and we get 



+ It/a 



(32) /(r«) = ^|f7coF(co)e-'"^'"*. 



We now let a converge towards zero and get, putting cc — ra 



+ 00 



(32*) /(ic) = ^|t7ü)7((ü)e-^^*' 



— 00 



The formulae (31) and (32*) constitute what is called the integral theorem of 

 Fourier. I shall call /(o)) the conjugate Fourier fund ion of J{x) and denote it with 

 f-\iü). We then have (introducing for the sake of symmetry a constant) 



+ 00 



(33) f'\^) = ^^\^dxf{x)e^'-\ 



- OD 

 + C0 



(34) fix) = I å^t • 



— 00 



The equation (33) (as well as (34)) may be regarded as an integral equation 

 for determining f{x) (or f~^). It is the first equation of this kind solved. If in 

 (33) f~''-{oi) is known the value of J(x) is found from (34). 



7. Solution of Schwarzschild. 



Let us consider a certain character of the stars. Then we have for its dis- 

 cussion, statistically, two equations (found in the preceeding lecture) of the form: 



+ C0 



(35) a{x)=ldç.à((.)^{x+p), 



— 00 



Lnnds Univ:s Årsskrift. N. F. Afd. 2. Bd 8. 3 



