VII. On the Theory of D finite Integrals. By W. H. L. Russell, Esq., B.A. 
Communicated hy A. Cayley, Esq., F.R.S. 
Received October 30, — Read November 23, 1854. 
I PROPOSE in the following paper to investigate some new methods for summing 
various kinds of series, including almost all of the more important which are met 
with in analysis, by means of definite integrals, and to apply the same to the evaltia- 
tion of a large number of definite integrals. In a paper which appeared in the Cam- 
bridge and Dublin Mathematical Journal for May 1854, I applied certain of these 
series to the integration of linear differential equations by means of definite integrals. 
Now Professor Boole has shown, in an admirable memoir which appeared in the 
Philosophical Transactions for the year 1844, that the methods which he has invented 
for the integration of linear differential equations in finite terms, lead to the summa- 
tion of numerous series of an exactly similar nature, whence it follows that the com- 
bination of his methods of summation with mine, will lead to the evaluation of a large 
number of definite integrals, as will be shown in this paper. It is hence evident that 
the discovery of other modes of summing these series by means of definite integrals 
must in all cases lead to the evaluation of new groups of definite integrals, as will 
also be shown in the following pages. I then point out that these investigations are 
equivalent to finding all the more important definite integrals whose values can be 
obtained in finite terms by the solution of linear differential equations with variable 
coefficients. Again, there are certain algebraical equations which can be solved at 
once by Lagrange’s series, and by common algebraical processes ; the summation 
of the former by means of definite integrals affords us a new class of results, which I 
next consider. A continental mathematician, M. Smaasen, has given, in a recent 
volume of Crelle’s Journal, certain methods of combining series together which 
give us the means of reducing various multiple integrals to single ones. The series 
hitherto considered are what have been denominated “factorial series”; but, lastly, 
I proceed to show that analogous processes extend to series of a very complicated 
nature and of an entirely different form, and for that purpose sum by means of 
definite integrals certain series whose values are obtained in finite terms in the 
‘Exercices des Mathematiques’ by means of the Residual Calculus. The total result 
will be the evaluation of an enormous number of definite integrals on an entirely 
new type, and the application of definite integrals to the summation of many intri- 
cate series. 
MDCCCLV. z 
