50 c. V. L. Charlier 



Let US omit the indices of the function R and write 



lijk = j {Ri — -^i)/i/2 '^i^ 



where now 



R^ = Rijk {u^, w^), 



This integral is to be taken over all values of n^, v^, w^; u.^, v.^, iv^ between 

 + 00 . If the variables and etc. change places, the value of the integral 

 remains unaltered. Hence we may write 



lijk = j {R^ — R^) U Â 4 dh 



and finally 



(105) lijK = i / + Ä2' —Rt — ^2) fi f% -^h db , 



whicii is symmetrical in and . 



To avoid confusion I write out the values of the four i?-f unctions 



R^ = Rijk (f/j, Wj, ) 



R.2 = Rijk («2' ^2 ) 



R^ = Rijk v^', u\') 

 R^'^ Rijk (wg', v^', w,') . 



It is formula (105) that is to be used for computing liß (and S/iß). 



Using the generating function (96), the value of Rijk may be written in the form 



Rijk («1, y^, w^) = Ri{u^) Rj(v^) Rk{w^). 



The values of Ri etc. are explicitly given in formula (69). 



38. Terms of the order zero. 



For i=j = h = 0 we get from (105) 



^ooo("i ' «"n ^'^1) = RM -^o("i) ^o('^) ^ 1 



so that 



R^ = 7?,, ^ R^ = 7?; = 1 



and 



(106) Vooo = Û. 



39. Terms of the first order. 



We have 



RiU^h ' ^\ ' ^1) = RMi) Roi^x) Roi'^i, 



