VOL. LXXIX.] PHILOSOPHICAL TRANSACTIONS. 5/5 



distance pot = v' (a^ + 2bi/ + ^'^)> and the function of the distance into the 

 particle xy = f \/{a'^ ^ 2hy + ^^) X i' = f (y) X y\ let this be denoted by Ix 



situated in the line vx, which resolve into 2 others nx = ^-Pt^jtW—. — ;^ 



px = -v/(« ± 26y + y^) 



situated in Che line ab, and In (in a direction parallel to ifj = . .^ ^ ^.' j! ,x ; 

 find the fluents of the fluxions " ^-^ x ^ • (J/ ^j^j qy x r . (y) f,Q^^\^^Qf^ between the 



PX ¥X 



values af and fb of the Ymefx = y, which suppose y and v respectively; through 

 the point p draw vy parallel toyZ> = y, and in the line py assume pw = v; com- 

 plete the parallelogram Tuzy; pz will be the force of the line ab on the point p. 

 Cor. If F : (y) varies as any power or root (2n) of the distance px =: -v/a* + 

 iby + y^), and n — 4- be an integer affirmative number or O, the fluents y and v 

 of both the fluxions can be found in finite algebraical terms of y; if « — -^ be 

 an integer negative number, both the fluents can be found in the above men- 

 tioned finite terms, together with the arc of a circle whose radius is -v/a* — b* 

 and tangent y ^ b, unless n — -f = — 1, in which case the fluent y involves 

 that circular arc, and also the logarithm of y"^ + iby -{• a^. If n — 4- denotes 

 a fraction whose denominator is 2, both the fluents can be expressed by the 

 finite terms together with the log. of 3/ + ^ -f- v (y^ ± Iby -f- a^). If the 

 fluents be given^ when n is a given quantity, and n — 4- not a whole affirmative 

 number, from them can be deduced the fluents of any fluxions resulting by in- 

 creasing or diminishing rz by a whole number, unless in the above-mentioned case 

 of n — 4- = — 1. If ^ == 0, and consequently the line vf is perpendicular to 

 the given line at, the fluent y will be expressed by the finite terms, unless n — 4. 

 = — 1, in which case it will be as 4- log. (2/^ -|- a^) when properly corrected. 

 These fluxions t and v may be transformed into others, whose variable quan- 

 tity is par = M the distance from p, by substituting in the fluxions fory and^ their 



respective values »/ (t^ — a^ -f Z?^) + b and _ a* + b^V and consequently for 



V{y^ ± 2by -{- a*) its value u. 



Prob. 2, fig. 7. Given the attraction of each of the parts of a given surface 

 in terms of their distance from a given point p, and an equation expressing the 

 relation between an absciss a/j = x, and its correspondent ordinates pm = 2/ of 

 the surface; to find the attraction of the surface on the given point p. 



First, by the preceding proposition find the attractions y and v of any ordinate 

 m p m' in the directions of the ordinate pm and of the line vp ; and from the 

 equation expressing the relation between the absciss and ordinates of the given 

 curve, find the absciss in terms of the ordinates {pm) =. -k : (j/), and thence 

 i = ^ : (y) x y and ^{a^ ± Isd a? -|- a?*) = (p' : (y), where pa = a\ and s = 

 cosine of the angle which the absciss Kp makes with the line pa; then find the 



fluents of the 3 fluxions i X y = ^r X y X ?> : (y),i ^ —^^^^^±-—^- ^^ >, 



