VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. SQJ 



assume the 2 equations before given (p 4- x'v) A = — vv and v"^ = p'k, and 

 since a is always given in terms of d and d, if f and f' be given in terms of d, 

 d', &c. the value of v X x' may be acquired by a simple algebraical equation ; 

 but if F and f' be not given, and consequently v not given, but v a given func- 

 tion of Vf and x' a given function of the above-mentioned distances ; then 

 substitute for v its value v^(F'fi) in the function v, and the fluxion of ^f'r for vv, 

 and there will result an equation involving d and f' and their fluxions, and p; but 

 if the forces tending to all the points but one are given in terms of the distance 

 D, or absciss or ordinate of the curve, and their fluxions ; then from f' can be 

 found p, and, vice vers^, from p can be found f', and consequently there results 

 a fluxional equation expressing the relation between p or p' and the distance d or 

 d', &c. orabsciss or ordinate, and their fluxions. From p and p', and con- 

 sequently V being found in terms of d, d', &c. can be deduced t = -. 



The same method may be applied, if some forces tend to an infinite distance, 

 that is, act parallel to themselves, and others tend to given points. 



Exam. Let the accelerating force be directly as the arc = a?, and the resistance 

 uniform = a ; then will (x — a) x = — vv, and consequently cc'^ — 2ax + b = 

 — v'^, let A be the arc, where the velocity = O ; then will the equation 



A^ — 2aA — 2'^ + 2ax = v"^, and the increment of the time ^ = - = 



V 



V(^^-2al^x^+2axy ^^^'^ '""^^^'^^ '^ I^^a >< ^""^ ^^ ^ ^^''^J^' «^ ^^ich the 



radius is a — a and cos. = x — a, where a is the distance of the point from 

 which the body begins to fall, and the lowest point of the curve; and the acce- 

 lerating force X — a is as the distance from a point (a) of a curve, of which the 

 distance from the lowest is a. 



Corol. The times of the body falling from any point of the curve to a will 

 be equal. 



Corol. The body on this hypothesis will either rest at the point a, or at the 

 lowest point, or any point between -f- a and — a ; for it may rest at any point, 

 where the resisting force is always equal to or greater than the accelerating force* 



Corol. Let n be the number of vibrations; then the distance of the arc, to 

 which it will ascend from the lowest point at n vibrations, will be a — 2na\ if 

 A — 2na be not greater than 2a, it will never pass the lowest point. 



Philosophical inquiries require some corrections, which do not enter into 

 mathematical calculus; for example, in some cases the calculus changes the 

 quantities from negative to affirmative, &c. when from philosophical considera- 

 tions they are not changed; and, vice vers^, they may be changed to affirmative,^ 

 &c. on philosophical considerations, when they are not changed from the cal- 

 culus: and also a body may stop, &c. from philosophical considerations, as in 

 the preceding example, when it does not follow from the algebraical calculus, &c. 

 It is further to be observed, that resistances are always to be taken affirmatively.. 



