382 Sir W. R. Hamilton on Definite and Multiple Integrals. 
(where D7"f¢ still denotes a descending and periodical series, 
analogous to and including those above marked (18) and (25),) 
consists in developing the algebraical expression (10), (for the case 
r=0, but with a corresponding development for the more general 
case,) according to descending powers of the symbol 1,, and retain- 
ing only those terms in which the exponent of that symbol is posi- 
tive or zero: which process gives the formula, 
ft= 11427071) = GIP — + gh =. 1; 
that is, by (5) and (6), 
iusto... Q-2m—-1 [—3] m [0] “"r0] —(n—2m—1) 4¢n—ImM—1 ; (28)! 
where the series may be written as if it were an infinite one, but 
the terms involving negative powers of ¢ have each a null co- 
efficient, and are in this question to be suppressed. 
For instance, I have arithmetically verified, at least for the case 
t= 10, that the two finite algebraical functions, 
¢ ye Bt 
ne eae te * aac i 
fet= 570 — 96 + a5e° (28) 
Py i) 37? 5 Vi 
t= 1990 384 +513~a01@ °° (8) 
express the values of the two following sums or differences of 
integrals, . 
Sef ale eae Ch. a, Vina: tomate See 
fLt=1/ft+J/ft; . . . . . . (27)! 
the calculations having been carried to several places of decimals, 
and the integrals I,°ft, 1,’f¢ having each been found as the dif- 
ference of two large numbers, 
Observatory of Trinity College, Dublin, 
September 29, 1857. 
[To be continued. | 
yet for large and positive values of t it is, arithmetically speaking, by much 
the most important portion of the whole: and accordingly I perceived 
(although I did not publish) it long ago, whereas it is only very lately that 
I have been led to combine with it the trigonometrical series, deduced by 
a sort of extension of Poisson’s analysis.—When I thus venture to speak 
of any results on this subject as being my own, it is with every deference 
to the superior knowledge of other Correspondents of this Magazine, who 
may be able to point out many anticipations of which I am not yet in- 
formed. The formule (27) (28) are perhaps those which have the best 
chance of being new. 
> dita 
(28) 
