332 Mr. W. H. L. Russell on the Summation of Series. [June 15, 



deduced from one in the paper of Professor Boole on the Theory of 

 Development : 



The method of the present paper is of course of far more general ap- 

 plication ; but I have said enough in it to explain the principle on which 

 such expansions must be conducted. 



IV. " On the Summation of Series." By W. H. L. RUSSELL, Esq., 

 A.B. Communicated by Professor STOKES, Sec. R.S. Received 

 May 13, 1865. 



In a Memoir published in the Philosophical Transactions for the year 1855, 

 I applied the Theory of Definite Integrals to the summation of many intri- 

 cate series. I have thought my researches on this subject might well be 

 terminated by the following paper, in which I have pointed out methods 

 for the summation of series of a far more complicated nature. 



I commence with some remarks intended to give clear conceptions of 

 the general method of calculation. 



In any series, 



. +a*u x +&c. 



Where a is less than unity, it is evident that we can sum the series by a 



definite integral when u x =J du U, \J*, ~U l and U being functions of u, and 

 the integral being taken between certain assigned limits. For it is mani- 

 fest that the quantity under the integral sign then becomes a geometrical 

 progression. 



Again, for a similar reason we can express by a definite integral the sum 

 of the series 



-f a*u x v x w x . . . + &c., 

 where 



u x =fdu 17,11*, v x = 

 Wje =fdwW^f x > &c. 

 Lastly, we can sum the series 



x w x . . . + &c. 

 by a definite integral when 



