PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. OOB 



Ex. To expand ,, — in terms of — • 



1 , .. .. /-« + i 



(1+^)- 



-» + i 



Generally .-;^7j= = (-1)-^ 



therefore, the coefficient of oc~''^ is 



IT (-i)-"/o~ /y/o " 



Hence, except w is of the form ^, where j9 is some odd number, the coefficient is 

 zero ; for l-n + \ is finite, and / o~ infinite. But if n = r + \, the coefficient is 



/-r/r + ^ f 



r + i 



/i /o {-iyi.2...r/± 



= (-1) 

 = (-1) 



+ (-1) 



1 2 .. 



, 1.3..5...(2r-l) 

 2.4.6... 2y 



.1.3.5... (2^--l) 



\/lT^ ' ' 2.4.6... 2y a;'-+l 

 + 



We now draw the memoir to a conclusion, trusting that it may be deemed 

 worthy of consideration, as well from the generality and completeness of the 

 methods exhibited, as from the simplicity with which they are demonstrated. 

 We could conceive more limited theorems than those which, occupy our last sec- 

 tion ; but as the subject is as yet little studied, and as no person appears to have 

 attempted the application of any such theorems hitherto, I hope partial defects 

 will be excused, and, if possible, remedied by those who enter into the subject. 



Edinburgh, December 2. 1839. 



VOL. XIV. PART II. , OR 



