NEWTON S PRINCIPIA. 23 



d 2 , d 3 , &c., instead, to express the differentials. In like 

 manner f for sum is used by the latter to express the 

 integral, and /by the former for the fluent. Although 

 the continental method of notation is now generally used, 

 and is on the whole most convenient, yet it has its inconve 

 nience, as the d is sometimes confounded with co-efficients 

 of the variable quantities ; it is in some respects, too, not 

 very consistent w r ith itself; as by making d x 2 mean the 

 square of the fluxion, or differential of x ; whereas it, 

 strictly speaking, appears to denote the differential of a: 2 . 

 There can be no doubt, however, which notation is the most 

 convenient in the extension of the system to the calculus of 

 variations, where the symbol is 8 ; for, although the varia 

 tion of a fluxion or differential may perhaps even more 

 conveniently be expressed by 8 x than by I d x, yet the 

 fluxion of a variation can with no convenience be expressed 



bv^-, or otherwise than by d%x. The expression of 

 * ox 



second fluxions undeveloped is also far less convenient 



by the Newtonian notation. Thus the fluxion of -^ is 

 J dx 



sometimes required to be expressed without developement, 

 as in the expression for the radius of curvature, where 

 it is often expedient not to develope it in the general 



equation, but to find -^ in terms of x or y before taking 



Ct 3C 



its fluxion ; yet nothing can be more clumsy than to place 

 a dot over the fraction, whereas d -r- IS perfectly con- 



venient. 



Several important considerations arise out of the nature 

 and origin of these infinitesimal quantities as we have 

 described them ; and to these considerations we must now 

 shortly advert, as they give the rules for finding the 



c 4 



