Application, the doctrine of uniform motion, that if v is the velocity 

 '*~~.'~~*' already acquired, and s the small space passed over du- 

 ring the next instant s=vt. 



In last case it appeared, that when a force continues 

 to act with the same intensity, and the mass remains 



the same,y= . Here, where the same intensity con- 

 's; 

 tinues only for an instant, j= 



(*)' 

 In a similar manner, since by last casef- , it fol- 



9B4 



2vv vv 



lows, that in tins case/= r= -r- 

 2s s 



Also since by last case/=^r-, hence in this case 



s = ~2ii = lT 



Farther, since s=v't, and consequently v= _, there- 

 it 



fore taking the fluxions of these quantities v=( 1, 

 substituting this instead of v in the second formula, we 

 get/= (-?-), which becamesf=4- when / constant. 



Hence, bringing together all the formulae necessary 

 in the present case, we have 



.. * . * 



t=vt, hence i, and -7-= v 



DYNAMICS. 293 



then a s will be the distance now from the point of Application. 



f . .' 

 =-^, hence/*=t*. 



t 



attraction, and a s\" multiplied by some constant quan- 

 tity, which we may call m, will be the force that now 

 acts on the body. It is evident that in any case m 

 will be known when we know the force at some given 

 distance. 



To discover ^he relations of ' and s, we will sub- 

 stitute the expression for the force, in the third formula, 

 where v and * occur. 



Hence m X ( *)" X s=w. 



Having now got an equation involving the fluxions of 

 v and s with known terms, we will, by taking the 

 fluents, get an equation involving v and s themselves 

 with known terms. 



Hence 



=(4- ), which becomes/=:-4- when i constant. 



These formulae may be employed either for finding 

 the relations o(s, v, and /, when you know the law of the 

 force, expressed according to some function of one of 

 them ; and, conversely, you have only to substitute the 

 given relation in the proper formula, and then integrate. 

 The law of the force is commonly expressed according to 

 some function of S ; the two following problems will 

 involve almost every thing that can be desired on this 

 subject. 



PROS. I. 



Suppose that the force varies as some given power 

 n of the distance from a point toward which the body 

 is moving, and which we may call the point of at- 

 traction, it is required to determine the relations of the 

 space passed over, the time elapsed, and the velocity ac- 

 quired, so as to enable us to infer one from another. 



Let a;=the distance of the body from the point of 

 attraction at the outset, 



=the space it has past over, 

 =the velocity acquired, 

 ' tin.- time elapsed, 



Now the constant quantity C must be such that t>=0 

 when s=0, hence C= 



" n 



Hence j_ ma _w_ 



n+l 2 



> ' ~~^+T~ ~2 



This applies whether n be positive or negative, whole 

 or fractional, unless when n= 1. In this case, the 

 formula is evidently absurd, and in order to have the 

 proper fluents, we must recur to the fluxional equation, 

 which becomes m X (a *)~' X s=w. 



Hence, taking the fluents, and employing L to de- 

 note hyperbolic logarithm, we get mxL (a ) + 



C-- 



-2' 



The constant quantity C must be such that t)=0 when 

 s=0, hence C=m x L, a. 



Consequently m X (L, a L, (a )J= 



OrmX L, = ~o"' 



Again, to discover the relations of * and t in any case, 

 we have only by the above formula to find the expres- 

 sion for v in terms of s, and substitute that expression 



instead of v in the fundamental equation =t- We will 



thus have a fluxional equation involving the fluxions of 

 * and t with known terms, and taking the fluents, we 

 will obtain an equation involving s and t themselves 

 with known terms. 



Thus, when n= 2, which is one of the most inte- 

 resting cases, then 



each term of the last fraction by a s. 

 Hence <= - 



</SS 



' Vas * 



Now the last part of this expression, viz 



' Vas 



7 XS +2-><^ 



X* 



the 



is evidently equal to -^ ,_,_ A 



V as s * V as 

 first term of which has for its fluent / as ^ and that 



