282 Mr. W. H. Russell on Theorems in the Calculus of Symbols. 



of the altered alignement, by the formation of continents where 

 seas had been, and the opening out of new seas for the habitation 

 of marine animals, thereby causing a gap in the geological records 

 so far as they have hitherto been discovered, than the direct 

 result of the changed conditions to which the inhabitants of 

 the seas, and even those of the lands, came to be subject on 

 account of the entire change in the alignement of the land over 

 the globe. 



[To be continued.] 



XXXIX. Theorems in the Calculus of Symbols. 

 By W. H. L. Russell, Esq., A.B.* 



THE following theorems in differentiation and integration 

 may be proved by means of the Calculus of Symbols. They 

 are a development of results which I gave in a Memoir published 

 in the Philosophical Transactions for the year 1861, and will, 

 I hope, be found interesting to mathematical readers. The 

 proof of these theorems will be suggested by the original 

 memoir. 





d„dld,dl d _ . . d Y du 



r]QO . . gZ) . ___ rp2n-r 1 . 



doc dx x* dx dx x 4 ' ' ' dx dx x n ~ l dx 



■\)r 



g»Tf d \y +I y u 



dx nr \\x r - l dxj J x^ n + ] 



d^ x 2n + 2 \ dx 2 / x 2n ~ 2 ' 



— n(n + 1)- I dx .xu + n. — ^— . (n + 1) . — - ^ I dx.x q u 



n—- 1 72—2 



the integral sign in the first member of the equation being 

 repeated (n + 1) times. 



* Communicated by the Author. 



