On the Extension of the Theorem o/Xeibnitz to Integration. 335 



it is affected by the electrical state of the atmosphere ? In fine, 

 what is the proportion of the photogenic rays at each hour of 

 the day, and at different points in space at a given moment ? 



L. On the Extension of the Theorem o/Xeibnitz to Integration. 

 By J. R. Young, Professor of Mathematics, Belfast*. 



THE following method of showing the applicability of the 

 theorem of Leibnitz to successive integration, though 

 .new to me, may possibly be found in some works on the inte- 

 gral calculus which I have not seen. I venture however to 

 give it here, chiefly because it suggests a consideration, in 

 reference to the " Calculus of Operations," which is perhaps 

 deserving of notice. 



If we put du x for udx, du 2 for u x dx, du 3 for u^dx, and so on, 

 and apply " integration by parts " to the differential vudx, we 

 shall have 



fvudx=vu x -f£u x 



dx 



dv S*d?v . 



= m ~ dlc U * + J dJ* u * d * 



dv d*v PdPv , 



dv d 2 v d 3 v d m ~ l v 



=OTl _ _„ 2+ _ U3 __ K4+ ^—. Ui , 



*n 



u m ax, 



> d m v 

 dx* 



the sign of this last quantity being opposite to that of the term 

 immediately preceding it, and which term, here regarded as the 

 ?wth term, may beany whatever in the series. Thus far there 

 is nothing new. 



Integrating again, and proceeding on the same principles, 

 omitting however the supplementary integral, which, as above, 

 should terminate each of the following rows, we shall have 

 dv d 2 v d?v 



a^ U3+ fa* U *~~!hP' 

 dv d?v d 3 v 



'dx Us+ dJ* u *~ d^ 

 d 9 v d 3 v 



dx~* U *~ d& 

 dH 

 dx? 



&c. &c. 

 * Communicated by the Author. 



/ 



*VUdx*=:VUi- — 1L+ XS M 4~ TO M 5 + 



+ —*"4- -1ZS U S + 



— !T3«5 + 



