878 Sir W. R. Hamilton on the Calculation of the Numerical 
be briefly stated as follows. Introducing the two definite in- 
tegrals, 
A;= a cos (2¢ vers @), 
To 
7 «eh es) 
i= =( * de sin (2¢ vers @), 
7 Jo | 
which give the following rigorous transformation of the expression 
(13)!, or of the function fz, 
ft=A;c0s2¢+B,sin2¢; . . . . (16)! 
and employing the limiting values, 
SF Wes 
lim,_,-#A,= = dx cos (x) = (2a) 4, 
To (16)" 
: 1 ee 
lim, HB .=— i dx sin (x?) = (27) ~*; | 
(which two last and well-known integrals have indeed been used 
by Poisson also,) I obtained (and, as I thought, more rapidly 
than by his method) the following approximate expression, equi- 
valent to that lately marked as (15)", for large, real, and positive 
values of ¢: 
fi=(wt)-*sin (t+ 7) 5. ot wp geal) 
which is sufficient to show that the large and positive roots of the 
transcendental equation, 
Tv 
(a 608 (27:cos'w) =0, 0 eel)! 
0 
are (as is known)* very nearly of the form, 
aa ae Mea a | 
where n is a large whole number. 
[6.] Poisson does not appear to have required, for the appli- 
cations which he wished to make, any more than the ¢wo con- 
* Jt must, I think, be a misprint, by which, in p. 353 of Poisson’s me- 
moir, the expression k=ir-++ a is given, instead of ir— 4 for the large 
roots of the transcendental equation y=0. It is remarkable, however, that 
-this error of sign, if it be such, does not appear to have had any influence 
on the correctness of the physical conclusions of the memoir: which, no 
doubt, arises from the circumstance that the number 7 is treated as infinite, 
in the applications. - 
