18 Proceedings of the Royal Irish Academy. 
If in the riglit-liand member of this equation we substitute for 
{i-n) {i + n+ l)D''Fi 
its value from equation (4), we obtain 
A„+i + («-^) {^ + n + l)A,, 
since u vanishes for ju. = 1, and also for ju, = - 1. 
Ey making /x = />t'= 1, Z,- is reduced to unity; but, on the hypo- 
thesis above, P» = P/ = 1. Hence we see that ao= 1. 
We have now, from equation (2), 
2i+ 1 
Hence, from (3), we obtain 
2 
Ai = - 'i{'i+ 1) 
A3 = {i - + 1) + 2) — , &c,. 
2i + 1' 
2i+ r 
and in general 
^ ^~ -^^^ 27Tl + ^) + ^ - 1) • • • - ^ + 
Substituting this value of A„ in equation (1), we get finally 
" (^• + ?^)(^■ + w-l)...(^•-^+l) ^ ^ - ^ ' 
The value of a^, as given by equation (5), does not seem to agree 
with the value which Laplace obtains for the numerical factor which 
he terms y. It is easy to show that this disagreement is only ap- 
parent. 
In the expansion of his coefficient, Z,-, Laplace takes unity as the 
coefficient of the highest power of /a in the function of [x multiplying 
cosw(<^ - <^'), 
