Values of a certain class of Multiple and Definite Integrals. 379 
stants, which he called A and B, of his descending series (15) ; 
although (as has been said) he showed how all the other constants 
of that series could be successively computed, from them, if it had 
been thought necessary or desirable to do so. In other words, he 
seems to have been content with assigning the values (15)!, and 
the formula (15)", as sufficient for the purpose which he had in 
view. In my own paper, already cited, I gave the general term 
of the descending series for ft, by assigning a formula, which 
(with one or two unimportant differences of notation) was the 
following : 
mar 
(wt)'ft=>,,_, [0]-"([—3]”)2(40)~cos (17 (18) 
As an example of the numerical approximation attainable 
hereby, when ¢ was a moderately large number, (not necessarily 
whole,) I assumed ¢=20; and found that sizty terms of the 
ultimately convergent, but initially divergent series (13), gave 
#(20) = 2 (“ao cos (40 cos ev) 
0 
—+7 447 387 896 709 949° 965 7957 
—7 447 387 396 709 949° 958 4289 
pe bi fV 2. BOE iho iSye nsoshter 04 0-84 ans AED) 
while only three terms of the ultimately divergent, but initially 
convergent series (18), sufficed to give almost exactly the same 
result, under the form, 
| 9  \cos86° 49! 52" 1 sin 86° 49! 52! 
f 20)=(1-s5r55) 7 a 
— 00069736 +0-0003936= +0-0073672, . . (19) 
[7.] The function f¢ becomes infinitely small, when ¢ becomes 
infinitely great, on account of the indefinite fluctuation which 
cos (2¢cos w) then undergoes, under the sign of integration in 
(13)'; so that we may write 
on RS A etre ge a Gh eee 
But it is by no means true that the value of this other series, 
es iy Pp (F) + ‘(i “cob a oleR) 
‘ 1 ePXI 5\1.2 
which may be expressed by the definite integral, 
Pi ot= 2 “do see o sin (2080), eel see OES) 
0 
