VOL. LXXVIII.] PHILOSOPHICAL TfiANSACTIONS. 387 



tion k' = O, which expresses the relation between the perpendicular sy = ?' and 



distance sp = d', of every curve which at equal distances has the same velocity 



and force tending to s; reduce the equations k' = O, d = v^(j?^ -f y"^) and wp' = 



yx — xy . g^ ^1^^^ ^ ^^^ ^/ ^ i^g exterminated, and there will result 



the same fluxional equation of the first order, expressing the relation between 

 x, y, and their fluxions, whatever may be the value of n. The general fluent of 

 this fluxional equation contains the relation between the absciss and ordinates of 

 all curves, which have the same force and velocity at the same distance as the 

 force and velocity in the given curve. 



Prop. 4. — 1. Let a body move in a given curve ph (fig. 7), of which the ve- 

 locity iy) at any point p is given: and let the forces/"',/"'', &c. tending to all 

 the given centres s", s'", &c. (except two s and s') be given; to find the forces 

 /and /' tending to the two points s and s'. — Draw a line po perpendicular to 

 the tangent z/pz/'; and from the given centres s, s', s", &c. draw lines s/ and s_y, 

 s7' and s'z/', s"/" and s*y\ &c. perpendicular to the lines po and yvy', &c.; then 



will - =/X - +/' X ^ +/'' X ^, + &c. where po is the radius of the 



PO "^ PS — "^ PS "^ PS ~~ 



circle having the same curvature as the curve in the point p, and 



^—^ = fx~ + fX — + &c. where a denotes the arc of the curve ph ; from 



A* *' PS — "^ PS ~ 



the data may be deduced all the quantities contained in the above-mentioned two 

 equations, except/ and /'; and consequently from the two given simple equa- 

 tions be deduced the forces sought /and /'. 



2. Let the velocity of the body moving in the given curve be supposed always 



uniform; then/ X ^ + J' X % +/" X %; + &c. = 0. 



» "^ PS — PS — "^ PS ~ 



Exam. Let the curve hp/ be an ellipse, and the two foci s and s' the centres 

 of forces; then will/X — =/' X — ,; but the angle spy = s'pz/', and conse- 

 quently^ = ^ and/ = /'; but since ll =/ x ^ +/ X ^ = 2/ X '-^, and 



i J HP S P J J ^ PO "^ SP ' "^ SP J '^ sp' 



SP' 



(V- X SP 



v = a, then will f = be the force tending to each focus. 



*^ 2sy X PO ° 



In these and the subsequent cases the lines py, pz/', pz/", &c. are to be taken 

 negatively or affirmatively, as they are situated on the same or different sides of 

 p; and in the same manner the lines p/, p/', p/", &c. are to be taken negatively 

 or affirmatively as they are situated on the same or different sides of the tangent 

 yvy', &c. 



3. Let the centres m, m', m", m"', &c. of forces, be points not situated in the 

 plane of the given curve hpi, &c. and the forces/"', /"", &c. tending to each 

 of the centres m'", m"", &c. (except three m, m', and m") be given ; to find the 

 forces/,/', and/" tending to those three points m, m', and m". — Draw ms, m's', 

 m"s", &c. perpendicular to the plane hpi, &c. from the above-mentioned points, 



3d 2 



