NEWTON S PRINCIPLE. 25 



and of -^9 or x m x y~ n = m z m ~ } y~ n dxn x m y~ n ~ l d y, 

 _mx m - l y n dx nx m y n - l dy 



Consistently with the same principles, we may deduce 

 this rule otherwise and more strictly. Let x + d x be 

 the quantity when increased by the differential. This 

 multiplied by itself, or its square when completed, is x* 

 + 2 xdx + (dxf\ but to have the mere increment or dif 

 ferential we must deduct x~ } and we must also reject (d #) 2 

 as evanescent compared with the function 2 x d x, which 

 leaves 2 x d x for the differential. So the cube is x 3 -f 

 3 x 1 d x + 3x(dx-*) + (dx) 3 &amp;gt; an( l rejecting, in like manner, 

 we have 3 x 1 dx ; and by the binomial theorem (x + d x) m is 

 x m +mx m - 1 d x, + 8tc. + (d x) m , of which only the second 

 term can upon the same principles be retained ; that is 

 - 1 dx: And the same rules apply to the differentials 



of surds; so that the differential of (x + yf is y 



* v x + y 



It also follows that the fluent or integral is a quan 

 tity such that, by taking its fluxion or differential 

 according to the foregoing principles, you obtain the 

 given fluxional or differential expression. Thus if we 

 have to integrate any quantity as x m d x, we divide by 

 m + 1, and increase the exponent by unity, and erase 



x m+l 

 the differential quantity ; so that - - is the integral 



required. But as every multiplication of any two quan 

 tities whatever gives a finite product, and every involution 

 a finite power, while we can only divide so as to obtain a 

 finite quotient, or extract so as to obtain a finite root, 

 where the dividend or the power operated upon happens 

 to be a perfect product or a perfect power ; so in like 



