Derivatives of the Product of t'^o Monome Functions. 265 

 whence taking the first primitive, and transposing 



but in the vei'y same manner 



-\-A-'2zU2z'") = -3c" 2=^- -lr( -3:" 3.^) ^^• 



whence by the substitution of these values 



_^_iuv) — _^,UV— _^U ^V + _';^.}l'ci!p— _4i"3.'^+&C* 



Taking again the first primitive of each member of this 

 equation, expanding the partial results, and adding, we find 



-2.("^) = _2=""-2 _3^« ut' + 3 _4,M2^^-* -5==" 3.^ + ^^• 



By treating this repeatedly in the same manner, we should 



arrive at expressions closely resembling the expansions of the 



negative powers of a binomial, and should obtain in general 



_™.(mi;)= UV r- , ,.-,uv-\-- — .„.U„V—i>LC. 



Ti^~^ ' —mz 1 — (wi+l)^ Iz 12 —(Tnx-\-1)z Iz 



where, if we change m into —n, we have 



(uv)= UV— V, ,,«,u + &c. 

 nz^ ' nz 1 (n— l)s Is 



exactly the expression which we before found for derivatives. 



In the investigation of the formula for primitives no no- 

 tice has been taken of the arbitrary constants which at each 

 step enter into the calculation, and it therefore becomes neces- 

 sary to examine whether the omission has not vitiated the re- 

 sult, it being a well known fact that the equality of two func- 

 tions does not necessarily imply that of their primitives. 



The arbitrary part of a first primitive being of the form a, 

 that of a second primitive must be az+b', of a third primi- 

 tive a£^ + bz + c', of a fourth a£ + b^ -\- cz-\-d\ and, in ge- 

 neral, of a primitive of the wth order the arbitrary part is 

 az^-' +bz^-'^ -ifC^-^ +&c.... +p2-j-y. 



If in the value of _i,(mu) we annex to the successive pri- 

 mitives of ti these arbitrary parts, the former value of this pri- 

 mitive will be increased by the following quantity : 



a{v- ^vz+ ^vz2- 3 J)/ + ^vz2 - &c. } 



-V b[— ,.u + 2.t'2 — 3^uz^ + ^S)f — &c.} 



+ <^ {+ 2z" - 3z'^- -^ Az"^-^'^'} 



+ d {— ^jv + ^jvz — &c. J 



+ c {+ ^y - &c.} 



+ &c. 

 N. S. Vol. G. No. Si. Oct. 1829. 2 M which 



