OF ARTS AND SCIENCES: JANUARY 14, 1873. 493 



» 



D (z°) D (x-) 



Since ^ is a function of z only, and ^ a function of x only, 



Dz •" Dx ■" 



while 2 and x are any two values of the independent variable (h being 

 arbitrary), the functional expression which constitutes either member 

 of equation [3] does not change its value with the independent vari- 

 able, hence " n — 2 x = c (a constant), [4] 



or D(x 2 ) = 2x Dx + cDx. [5] 



To determine the unknown constant c, we differentiate, by equation 



[5], the identity 



(m x) 2 == m 2 x 2 , 



obtaining 2mx-mDx-{-cmDx = m 2 2xDx-\-m 2 cDx, 

 or c m (1 — m) Dx = 0. 



c = 0, 

 since m and D x have ai'bitrary values, 



JD(x 2 ) = 2xDx. [a] 



Equation [a] may also be deduced from [3] by the following method : — 



In equation [3] not only are x and z entirely independent, since h is 



D (z 2 ) 

 arbitrary, but D x and D z are no longer restricted, for J. will not 



change its value if we suppose Dz to have apy value greater or less 

 than D x. We may therefore put 



z=z mx. 



Introducing this value in [3], we obtain 



D(m*x*) ' D(?*) 



D (mx) D x 



ROD. 9 #Q 2 ) o 



or m ^ — 2 mx= S: — 2 x; 



Dx Dx 



Whence, since m is arbitrary, 



Dx 



