Values of a certain Class of Multiple and Definite Integrals. 381 
which ought, if the present theory be correct, to be nearly equal 
to the definite integral, —J,ft, for the case where t=20. In 
other words, I am thus led to expect, after adding the constant 
term i, that the value of the connected integral, 
T 
i= n= “do secwsin (40cosw), . . + (26)! 
0 
must be nearly equal to the following number, 
0562868. -eieeiix Har eee)? 
And accordingly, when this last integral (26)! is developed by 
means of the ascending series (21), I find that the sum of the first 
sixty terms (beyond which it would be useless for the present 
purpose to go) gives, as the small difference of two large but 
nearly equal numbers, (which are themselves of interest, as re- 
presenting certain other definite integrals,) the value : 
a) * do sec w sin (40 cos @) = 
0 
43 772 428 770 679 800:537 7058 
—3 772 428 770 679 799-974 8177 
2 MOE? aa Te 24a seals “EL Rey 
which can scarcely (as I estimate) be wrong in its last figure, 
the calculation having been pushed to more decimals than are 
here set down; and which exhibits as close an agreement as 
could be desired with the result (26)! of an entirely different 
method. 
[9.] It must however be stated, that in extending the method 
thus exemplified to higher orders of integrals, the development 
denoted by D; “ft, or the definite and multiple integral (—J i) Ae 
to which it is equivalent, comes to be corrected, in passing to the 
other integral I? ft, not by a constant term, such as 3, but by a 
finite algebraical function, which I shall here call f,¢, and of 
which I happened to perceive the existence and the law, while 
pursuing some unpublished researches respecting vibration, a 
considerable time ago. Lest anything should prevent me from 
soon submitting a continuation of the present little paper, (for I 
wish to write on one or two other subjects,) let me at least be 
permitted now to mention, that the spirit of the process alluded 
to, for determining this finite and algebraical correction*, 
Pf—(—I)ft=Uft—Di"ft=ft, - - - (27) 
* Although this algebraical part, f,,t, of the multiple integral I7ft, is here 
spoken of as a correction of the periodical part, denoted above by D; "ft, 
