396 PHILOSOPHICAL TRANSACTIONS. [anNO I788. 



1 . 2. Let e* = x' and x = ^, which is supposed to correspond nearly to the 

 state of our atmosphere; then will v^ = — 2 X e X / e Xx.r = 



- 2 X e-/"^^'ye>'"~X/'i=~2e-^"'"* - (y^e^-' + ' X b.v + a), e being 

 the number, whose hyp. log. is 1, and h and a quantities to be assumed accord- 

 ing to the conditions of the problem. 



1. 3. Let x = x'; then it becomes xi = ~~ *" and t = —:r^, r. 



I + av X (1 + a\) 



2. Let X be an homogeneous function of one dimension of x, that is, = ax^ 

 and V a similar function of n dimensions of v, that is = b%f, and x' a similar 

 function of r dimensions of x and v, and n -f- ^' = 1 ; then by substituting zx 

 and its fluxion for v and its fluxion, can be found the fluent of the fluxional equa- 

 tion (x -|- avx') i= — vv, and consequently the velocity and time by the quad- 

 rature of curves in terms of the space; and in like manner of many other cases. 



3. Fig. 6. Let a body, moving in a given curve, be acted on at any point p by a 

 force y tending to a given point s, and resisted by a medium proportional to v, 

 a function of its velocity multiplied into its density x', a function of the distance 

 sp = D ; to find its velocity, time, and distance from the given point s in terms 



of each other. Let f =/ X — the force in the direction of the tangent py, 

 and consequently (p -^- vx') k = — w, and i;^ = 4- c Xfi where a is the incre- 

 ment of the arc, and c the chord of curvature in the direction sp ; but since 

 the curve is given, the chord of curvature may be deduced from the distance, &c. 

 and the increment a of the arc from a function of the distance multiplied into 

 the increment of the distance; then, if /" or v he a given function of the dis- 

 tance, the other may be deduced from it, and consequently — »{, = (p : (d) X 

 i, will be a given function of the distance d multiplied into d, whence we have 



f : (d) X i) = i) (/X h x'v) divide by d, and there results an algebraical 



equation, from which v X x' may be found. 



If neither v nor f be given, reduce the 2 equations (/ x h vx') a = •— 



tv and z;* = 4- c/, into 1, so as to exterminate either/ or v and its fluxions; and 

 ihere results an equation expressing the relation between the other v orf and d 

 and their fluxions: from the velocity given in terms of d may be deduced the 



time from the equation t = -. 



3. 2. If the body be acted on by forces tending to more points s, s', s'', s"\ &c. 

 in the same plane ; resolve each of the forces into 2 ; one in the direction of the 

 tangent, and the other perpendicular to it ; let the sum of the forces in the 

 direction of the tangent be f ; and in the direction perpendicular to it be p' ; 

 and 2r the diameter of curvature at the point p, which will be given in terms of 

 the distances from 2 points, or of an absciss and ordinate, and their fluxions, &c.; 



