Mr. J. O'Kinealy on a New Formula in Definite Integrals, 295 



conformity of position, but also a common origin. The develop- 

 ment of the variation of Hamilton's integral has the peculiarity 

 that it leads simultaneously to the differential and integral equa- 

 tions of mechanical problems. This remarkable fact gives to 

 the principles of Energy and Action a common origin in the 

 general equation of motion ; and by this the latter becomes the 

 connecting band for the two propositions of the mechanical 

 theory of heat. 

 Ziirich, April 1874. 



XLI. On a New Formula in Definite Integrals. 



To the Editors of the Philosophical Magazine and Journal. 



1 1 Elysium Row, Calcutta, 

 GentlemEx\% August 2, 1874. 



I SEE in the July Number of your Magazine two new for- 

 mulae in definite integrals of some importance are given by 

 Mr. Glaisher. The integrals admit of a direct general solution 

 without using the identity on the right-hand side of the equa- 

 tion, 



The portion to the left is evidently equal to 

 _1 



or putting E = e Da ; 



rdr^ o=wsay - 



From this 



And taking the limits a. and 0, the value for this particular case 

 is 



E >. ry a =-.a.,. 



The second theorem is obtainable in the same way. It is 

 * udx 



Jdn 

 iT^.(i+E^) ,f/ ° 



= (1~E)- 1 . {tan-^-EHan-'E^K. 



J 



