54 A Discourse on the Theory of Fluxions. 



accelerated velocity, and let the increments Gs Pm be gen- 

 erated in the same time; let also Pv be the increment that 

 would have 



F G s 



1 1 K 



A P «i Z 



l—l-i 



been generated in the same time, if the velocity at P had 

 been continued uniform ; then by Prop. I. the fluxions of 

 FK, AZ, at the points G and P, will be represented by Gs 

 and Pv. Let V be the velocity at P, or the velocity with 

 virhich Fv is described, and let r be the increase of velocity 

 from P to m ; then the velocity at m will be V+r, and V7n is 

 the increment which is described in consequence of the in- 

 crease r of velocity since the describing point left P. Now 

 let Y-\-w be the uniform velocity with which Pm would be 

 described in the same time that Pv and Pm are described, as 

 before mentioned; then it is manifest, that this uniform ve- 

 locity must be between the velocities at P and m, that is, V 

 -{-w is greater than V and less than V+r, or iv is greater 

 than o and less than r. Also, since the spaces described in 

 the same time are as the velocities, V : V+^-'^P^ • ^^n. 

 Now in every state of these increments,V : Y-\-iv: .Fv : Pm; 

 and by contiually diminishing the time, and consequently 

 the increments, we diminish r and tv, but V remains con- 

 stant; it is manifest therefore that the ratio of V : V+w;, 

 and consequently that of 7v ', Pm, continually approaches 

 towards a ratio of equality, and when the time, and con- 

 sequently the increments, become actually =0, then r=0 ; 

 consequently w=0:, therefore the limit of the ratio of P 

 V : Pm becomes that of V : V, a ratio of equality. Hence 

 the limit of the ratio of Gs : Pm is the same as the limit 

 of the ratio of Gs : Pr, or it is Gs ; Fv, that ratio being 

 constant." 



From the foregoing reasoning it is manifest, that the limi- 

 ting ratio of the increments expresses accurately the rate of 

 increase in the fluent at any assigned point in its generation. 

 An example in geometrical quantities of two dimensions may 

 be derived from the square, in which the two generating lines 

 constitute the limit. The ultimate ratio, then, expressed by 

 its usual representatives will be '2x : 1 , or, combining the 



