26* Mr. Sang u?i a Property of the Primitives and 



Taking the first derivative of each member of this equation, 

 we obtain 



^J^uv) = ij-Ci,"^) + i.('<iJ'); t^ut, by the same theorem 



and IjC^Iz'^)" u" i '^ + "oT* wherefore 



Applying the same method repeatedly to this equation we ob- 

 tain 



3,(»i v) •=. ^u u + 3 ^xi j.r; + 3 j j/ ^v + u ^v ; 



&c. &c. 



Where the order of the subponents is exactly similar to that 

 of the exponents of the integer positive powers of a binomial, 

 and where the numerical coefficients are formed in the same 

 manner. Hence in general, 7i being a whole positive number, 



nz(«^) = „."«+ f (— »=" lz^+ T ^(n-2)^"2." +&*^- 



And it is my object to show that the same theorem applies 

 when n is a negative integer. But before proceeding to that 

 part of my subject, I may notice a very simple extension of 

 the above theorem. 



As such expressions as ^ ^ '3 i T' 1 2" — 7 ^^^ ^^ ^'^'"^ ^^^' 

 quent occurrence in investigations of this kind, they may con- 

 veniently be denoted by rji and z' ; this premised, the above 



equation, dividing each side by 1.2.3 n, becomes 



nz'jJ! = «.«^ + («-l).!il.^+ (n-2>«2.-",+ &C. 



Extending the same reasoning to a product of three functions, 

 we have 



in which expressions a, b, c, &c. must receive, combined in all 

 possible ways, every integer value from to n. 



Resuming the original equation i2(U?;) = i-Uu + U i-t', 



and supposing U = _ i^u, and consequently isU = u, we have 



whence 



