494 PROCEEDINGS OF THE AMERICAN ACADEMY 



The process used in the case of those functions whose differentials 

 it is desirable to deduce independently is, in each instance, similar to 

 that used above in the case of D (or), and may be thus described : : — 



We assume a new variable z, connected with a; by a relation admit- 

 ting of a comparison of Dz and Dx, and at the same time such, 

 that D (f,z) and D (f,x) may likewise be compared; in other words, 

 such that the relation between z and x, and also between f, z andy, x, 

 can be differentiated without the introduction of unknown differentials, 

 except those denoted by D (f,x) and D (f z). 



■d j- • • .1 .- J> (/> *) J D (/> Z ) J J • 



By division, the ratios — £.- — - and — ~~ — - are introduced in a sin- 



J Dx D z 



gle equation. The arbitrary constant introduced in the assumed relation 

 between z and x is then eliminated, and the equation reduced to such 

 a form that one member is apparently a function of z, and the other 

 of x. This last process we call the separation of the variables. 



As x and z may denote any two values of the independent variable, 

 the apparent functions mentioned above will necessarily be identical in 

 form, and (since they constitute the two members of an equation) iden- 

 tical also in value. This value will be constant, since either member 

 of the equation is a functional expression, which does not change its 

 value with x. 



The determination of this constant is then effected by the differentia- 

 tion of some algebraic identity. 



The Differential of the Product. 



From the above expression for D (x 2 ), we obtain the Differential of 

 the Product, thus : — 



(x-\- y y = x 2 -\-2xy-\-y\ 



2(x + y)(Dx + Dy) = 2xDx + 2D(xy) + 2yDy, 



or xDx-\-yT>x-\-x Dy-\-yDy = xDx-\-D (xy) -\-y Dy, 



D (xy)=y Dx-\-x Dy.* [b] 



From this result the Differential of the Qiiotient is easily obtained. 



'The Differential of the Power. 

 Let z = rx, then Dz = r Dx, [1] 



and z m = r m x m , then D (z m ) = r m D (x m ), [2] 



* This method of deriving D (xy) from (Dx 2 ) is taken from Vince's Fluxions. 



