Tarleton — On Laplace's Coefficients. 19 
and puts 1 - /x^, instead of ^j? -1 or u. In our mode of writing 
Zj, which, is that adopted by Williamson, Differential Calculus, 
cosn{4> - cfi') is multiplied by a function of fx, which is expressed 
by B'^Fi, or 
The coefficient of the highest power of /a in this is plainly 
2t{2i- l)(2^-2)...(^•-?^+ 1) 
which may be written 
|_2e 
2i{2i -2). . ,2.^i-n' 
and this last is manifestly 
(2^'- 1) (2^-3)... 3.1 
- n 
Hence we see that 
(1 . 3.5 . . . (2^-l) 
>...(2^-l))2 , 
whence 
_ 2 | 1.3...(2^-1) |2 l^-n ^ 2 { 1 . 3 . . . (2/ - 1))^ 
I L** ~ ^ + ^ " u' - ^ . + 
This last expression for y is at once reducible to that given by Laplace, 
viz. : — 
1 . 3 . 5 . . . (2« - 1))2 , ,(t-n + l) 
{i+ l){i + 2)...{i + n)' 
C 2 
