380 Sir W. R. Hamilton on the Calculation of the Numerical 
is infinitesimal when ¢ is infinite. On the contrary, by making 
2\-3 
2t cos @=2, do sceo= — “(1-2 sion see 
x 
4p 
the integral (21)! becomes, at the limit in question, 
1(%dz. ‘ r 
Fo o= — : Se: cies bn tit ae (21) 
Accordingly I verified, many years ago, that the series (21) takes 
nearly this constant value, 3, when ¢ is a large and positive 
number. But I have lately been led to inquire what is the cor- 
rection to be applied to this approximate value, in order to obtain 
a more accurate numerical estimate of the function F,o¢, or of 
the integral I,,/#, when ¢ is large. In other words, having here, 
by (3) and (21)", the rigorous relation, 
Bt Lae ee Ce) 
I wished to evaluate, at least approximately, this other definite 
integral, —J, ft, for large and positive values of ¢. -And the 
result to which I have arrived may be considered to be a very 
simple one; namely, that 
= FED, pts IF ce ae) 
where D>" ft is a development analogous to the series (18), and 
reproduces that: series, when the operation D; is performed. 
[8.] As an example, it may be sufficient here to observe that 
if we thus operate by D; on the function, 
‘ — |—- — ——— oor eS Oe”, 
ft ( PUL 2a at a a 
and suppress ¢~# in the result, we are led to this other function 
of ¢, 
cos(2¢— =) sin (2e- vy, 
Dift=(1- pe Fey eee Lf. 
RPL art QV ort 
which coincides, so far as it has been developed, with the ex- 
pression (18) for ft: so that we may write, as at least approxi- 
mately true, the equation 
JSD ft... + eile eee 
Substituting the value 20 for ¢, in order to obtain an arith- 
metical comparison of results, we find, 
, 129 \sin 86° 49! 52" cos 86° 49! 52" 
F 20)=(1- spi500) Tee 
= +0:062942—0:000054= +0°062888; . . (26) 
(25)! 
