an Expression for the Numbers of Bernoulli. 365 



A 2/ ,= (-l)'- 1 2(2,)- 2 "2 (l . ) 7i-^ ) . . (25.) 



as at least approximately true, when the number^? is large. But 

 in fact the expression (25.) is rigorous for all whole values of p 

 greater than ; as we shall see by deducing from it an ana- 

 logous expression for a Bernoullian number, and comparing 

 this with known results. 

 S. The development 



_L_ + | U h- \ + B,^ - B 3T -^ + ftftj (26.) 



being compared with that marked (20.), gives, for the pth 

 Bernoullian number, the known expression 



B 2 „_ 1 = (-l) i; - 1 1.2.3.4...2 i ;A 2i ,;. . (27.) 



and therefore, rigorously, by the equation (19.) of the present 

 paper, 



B 



(-l) p ~ 1 1.2...2p 



(28.) 



/*" d / sin x \ 2p cos {2 p x - x) m 

 J \ X ) cos X 



a formula which is believed to be new. Treating the definite 

 integral which it involves by the process just now used, we 

 necessarily obtain the same result as if we combine at once 

 the equations (25.) and (27.) • We find, therefore, in this 

 manner, that the equation 



(in which, by the notation here employed, 



is at least nearly true, when p is a large number ; but Euler 

 has shown, in his Jnstitutiones Calculi Differ entialis (vol. i. 

 cap. v. p. 339. ed. 1787), that this equation (29.) is rigorous, 

 each member being the coefficient of u p in the development 



of — (1 — w u cot 7r u). [See also Professor De Morgan's 



Treatise on the Diff. and Int. Calc, * Library of Useful 

 Knowledge/ part xix. p. 581.] The analysis of the present 

 paper is therefore not only verified generally, but also the 

 modifications which were made in the form of that definite 

 integral which entered into our rigorous expressions (19.) and 



