TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 31 



Multiplying member to member and summing, we get 



J \y-V J \ 7 / (7-1)7 7 ^ 7+1 ' 

 We write this for a+1, /3+1, y+1, in the places of a, /3, y, and get 



( c > f a + 1 >/ 3 ± 1 /? a+l,/8+l = (q + lX<Q+l) ,/q + 2,/3+2\ 

 7 7 ~ 7+1 7(7+1) •'V 7+2 / 



By the combination (c). - (a) - (6) we get, after reduction of the terms in the first 

 member, 



q,/3 ,q + l,/8 + l = (q + l)(/3+l) ,f«+^+2 

 7 7 ' 7+1 7(7 + 1) A v + 2 



7(7+1) ■> \ y+! 

 7 7 \ 7+1 



y \ y + 



The two first terms in the second member are 



q + ir/3+1 f fa±%^±^\_ f (a±2 JL § + l^ 

 7 L7 + 1 7 V 7 + 2 / 7 V 7 + I J]' 



But if in equation (a) we transpose the first term on the left to the second member, and 

 write the whole equation by augmenting every letter by unity (a+1 into a+2,/3, &c), 



then we see that the factor of in the preceding expression will be 



J \ 7+1 / 



7+: 



Collecting now all the terms containing this last function, and passing them into the 

 second member of the preceding equation, we get 



7-ft-q-l f(a±}Jl±}\ 



7 7 V jTT~~J- 



7 \ 7- 



Writing this equation successively for a+1, /3+1, y+1, for a+2, /3 + 2, y+2, &c, up to 

 a +ti, fi+ti, y+ti ; multiplying all the results, member to member, and dividing out the 

 factors which both products will have in common, we get 



f fa,0\_ (y-/3-a-l)^- 1 { ((^±t})Jj3±ti}) 



J \ 7 J' yi/+l " J\ (y + tj J' 



The ratio in the upper factorial is = — 1, because 



(7+l)-(/3 + l)-(a + l)-l = (7-/3-a-l)-l 



