Values of a certain class of Multiple and Definite Integrals. 377 
which admits of being summed as follows, 
ji= 2 *do.cos (2te0s.0); oy eae hs cits 
0 
the function ft being thus equal to a celebrated definite integral, 
which is important in the mathematical theory of heat, and has 
been treated by Fourier and by Poisson. 
[4.] It was pointed out* by the great analyst last named, 
that if there were written the equation, 
y= {cos (cos) do, ig. ei 5 Pees) 
0 
so that, in our recent notation, 
y=nf(), SON hind, ate. batetres ae (TAP 
then for large, real, and positive values of k, the function yWk 
might be developed in a series of the form, 
! " 
yVk=(A+ 242 4+ &e.) cos k 
1 " 
+ (B+ 243 + &e.) sink; . » (15) 
where a certain differential equation of the second order, which 
yk was obliged to satisfy, was proved to be sufficient for the 
successive deduction of as many of the other constant coefticients, 
A!, A",.. and B!, B", .. of the series, as might be desired, through 
an assigned system of equations of condition, after the two first 
constants, A and B, were determined; and certain processes of 
definite integration gave for them the following values, 
DSS ER FRG OAS ia +e) 
so that when & is very large, we have nearly, as Poisson showed, 
yVk=(cosk+sink)/m . . . (15)! 
[5.] In my own paper on Fluctuating Functions+, I sug- 
gested a different process for arriving at this important formula 
of approximation, (15)", which, with some slight variation, may 
* In his Second Memoir on the Distribution of Heat in Solid Bodies, 
Journal de V Ecole Polytechnique, tome xii. cahier 19, Paris, 1823, pages 
349, &e. 
+ In the Transactions of the Royal Irish Academy, vol. xix. part 2, 
p- 313; Dublin, 1843. Several copies of the paper alluded to were distri- 
buted at Manchester in 1842, during the Meeting of the British Association 
for that year: one was accepted by the great Jacobi. 
