6 0] 
375 
60. 
ON THE EXPANSION OF INTEGRAL FUNCTIONS IN A SERIES 
OF LAPLACE’S COEFFICIENTS. 
[From the Cambridge and Dublin Mathematical Journal, vol. III. (1848), pp. 120, 121.] 
Suppose 
— &Qs 4" &iQs—i 4" 
(1). 
where, as usual, Q 0 , Q 1} &c. are the coefficients of the successive powers of p in 
(1 — 2pp + p 2 ) -i . Assume 
where A refers to C; then expanding this expression, first in powers of g, and then 
in a series of terms of the form a/(14-A 2 ).Q, and comparing these with the preceding 
values of S, 
o-s-r = \/(l 4- A 2 ) .A r .C 
A = I (1 - V(1 - 8% V( 1 + A y = ¡1 - vU - 8*))» 
(4). 
Then