266 Mr. Sang ofi the Product offmo Monome Functions. 



which at first sight appears to involve not only every positive 

 integer power of z, but even, since v is any function of ;:, its 

 fractional powers. 



We have, however, <^{z + dz) = fz+ y^<pzd z+ izf z d z^ + 



&c. in which if we make dz =—z, and change 'P z into u, 

 we obtain 



(t> {z—z) = <J> o = u— ^yz + 2z^^^~ 2z^^ + ^'^• 



and by changing i^z into ^v 



^J>{Z — Z) = i,<I> = ^J)— 2z^2+ 2~yZ.~ ^^• 



Now all such expressions as 4>o being constant quantities, it 

 follows that, by the addition of an arbitrary constant at each 

 integration of u, a constant quantity only has been annexed to 

 the first primitive o(ui\ and theretbre the superior primitives 

 cannot be afiected by powers of z so high as the order of the 

 primitive. 



Having thus shown that the omission of the arbitrary parts 

 has not rendered our result less general, I may proceed to il- 

 lustrate its utility by a single example. 



Let the 7th primitive of s^. sin z be required. Here making 

 s^ = V, sin z = u, we have 



_^^(z^ sin z) = _- sin z.2^—\ _g,sin z.Sz' + j- | _g,sin z . 6z 

 ~f I f -lO"''^^ ^ • ^' ^^> since _»„(sin z) = cos z; _g^ sins 

 = sin 2 ; _Q.sin z = — cos z ; and _jQ_sin z = — sin z 



_-.(2^sin z) = cos 2. 2^—21 sin 2 .2- — 168 cos 2. 2 + 504 sins. 



And it appears from this example that, of the functions u 

 and V, whenever the primitives of the one are known ai.d the 

 number of derivatives of the other finite, the integral of the 

 product will be obtained in finite terms ; in all other cases the 

 result will be in the form of an interrainate series. It may 

 also be noticed that two distinct expressions may be obtained, 

 the one combining the primitives of u with the derivatives of 

 V, and the other the derivatives of w with the primitives of u. 



When the expression to be integrated cannot be separated 

 into two factors, it may have the multiplier 2° supplied, which 

 artifice would give 



, „\ m 7)1 wi+1 m VI + m4-2 



1 ^ z"» ,2»»+l sm+2 J 



~ '.2....(m-l) IHT •''^ ~ ^iT+l l^"^ "in^ 2z^~ S 



which 



