VOL. LXXVIII.] PHILOSOPHICAL TANSACTIONS. 3Q5^ 



from the preceding principles; for v = py, the velocity at p, is inversely as the 

 perpendicular sy = sm let fall from the centre of force on the tangent; but sa^ : 



2sr X PA :: velocity py as — = — : p/ the velocity at m; whence p/^ (the square 



r i-u 1 A 1. \ *sp* X pa' .^ 2 1 • 1 • 4sp2 X PA» ^^ 1 

 or the velocity at m) = — X p"x , which vanes as j — X — : = 



•' ' Sa4 ' Bi» BUT* 



4pa* 4sa* — 4sA X X 



SY SM 

 * X PA* 

 SA^ ^->, - - , — . ^^4 ^, ^^3 



, and consequently as , the same as above. 



SA' X SM sa' X X 



2. Fig. 1 1 . If any number of forces act on a body at p in any given direc- 

 tions parallel, or tending to given points; resolve all the forces into 2 others; 1 

 in a given direction sm, and the other in a direction pm perpendicular to it, of which 

 let p be the sum of the forces resulting in the direction mms, and f the sum 

 of the forces resulting in the direction pm; resolve the velocity v of the 

 body at p, which is in the direction of the tangent py, into 2 others v' and V, 

 one in the direction parallel to the line sm, and the other perpendicular to it: 

 in the same manner resolve the velocity v of the body at p, which is in the 

 direction of the tangent pi/, into 2 others v' and ?;'', one in the direction parallel 

 to the line sm, and the other perpendicular to it : then if the velocity of the 

 body moving in the right line sm at m be v', and it be constantly acted on by a 

 force =: p, the velocity of the body at m will bei;': and if the body move from p 

 in a direction perpendicular to sm with a velocity as v", and be always acted on 

 by a force y, the velocity at the distance pm — pm will be v'\ 



Cor. From the forces given and the velocities in the above-mentioned direc- 

 tions at the point p, can be deduced the velocities in the same directions at the 

 point j&, and consequently the tangent to the curve at the point/;. 



Prop. g. — 1. Let the resistance of a body, moving in a right line, be as any 

 function v of the velocity i ; then will ; = -, i = ——; where t, v, and i, 



denote the increments of time, velocity, and space; their fluents properly cor- 

 rected will give the time and space in terms of the velocity. 



2. Let a body move in a right line, and be acted on by an accelerating force in 

 that line, which varies as any function x of the distance x from a given point; 

 and resisted by a force which is as any function v of the velocity v into its density 

 x', which varies also as a function of a? and v, then will (x -|- avx) i = — vv, 

 from its fluent x can be found in terms of v, or v in terms of x ; and thence i 

 = — „ of which the fluent properly corrected gives the time. 



Exam. 1. Let v = v^, and x' a function of x; that is, let the resistance be 

 as the square of the velocity and density; whence (x + av'^x*) x = — w, of which 

 equation the fluential will be 

 e X -IrV =z — / e X x.r 4- a, and t = / — r-, 77 — : 



^ * J ... J V-Ce-i^x'x xC/e>«' Xxi + A)) 



4- B, where a and b are invariable quantities to be assumed according to the 

 conditions of the problem. 



a E 2 



