VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. SSQ 



velocity in any point p. a denotes the arc of the curve, and d, d', d^, &c. the 

 respective distances of the body from the centres of forces. 



CoroL The increment of the time of describing any arc of the above-men- 

 tioned curve will be as the increment of the arc = a divided by the velocity 

 found above, and consequently the time itself will be as the fluent of it properly 

 corrected. 



Prop. 6. — 1. Let a body move in any curve, and be acted on by forces tending 

 to any given points, s, s', s''', s''", &c. ; all of which, except the force/ tending 

 to the point s, let be given; to find/ the force tending to s.-^Let sy, s'y\ s"y", 

 &c. be perpendicular to the tangent pj/ of the curve at the point p ; resolve the 

 forces//,/',/'", &c. tending to s, s', s'', s% &c. respectively into two forces, 

 of which one acts perpendicular to py, the other, s/, s7', s"l", &c. perpendicular 

 to po, which is perpendicular to py; let po be radius of the circle of the same 

 curvature as the curve, and v the velocity of the body at the point p ; then will 



i =/X 2 ± /' X '-^ ± /'' X %±f" X^X+ &c. and -i=fX '-^ 



TO -^ SP-' SP-' SP-' Sj/ A ''SP 



+ fX^ + f" X^-f± r X ^-r ± &c. : for '^-^iI2 = ^ chord of the circle 

 of curvature, which passes through s, write c; and for po x (±/' X ^-~ ± f'^ 

 X ^r- + &c.) substitute h, and for — write b ; and for + /' X ^ + f" X ^' 



S P — ' SP — -^ S P ' ^ g"p 



-j- &c. substitute d, and the two preceding equations become v^ ■= f X c -f h 

 and — vi) = (B/-f- d) a, where A denotes as before the increment of the arc of 

 , the curve; from the first equation w = ^ ^"^ = — (b/a + da) and con- 

 sequently c/ -|- (c + 2ba)/+ h + 2dA = O, from which fluxional equation 

 may be deduced the force / tending to the centre (s) = — c~' X 



_ /»2ba /^Ba 



g J c ^ / ^jj _j_ 2d A) X e*^ ^ ; where e is the number whose hyp. log. = I. 

 CoroL Fig. 3. 'LQ.\.f\f\f'\ &c. be each = O, then will d = O, h = O, and 



/*2ba 



consequently y (2 DA -|- h) X e*^ ^ = const = a, and/= _ a X c"' X 



_ /»2ba /*2sji 



e^ ^ = = ^= — rtc ' X e^ ^ ; whence/= — as is 



C X Pj/ sy '' SJ/*^ X c 



generally known, where a denotes an invariable quantity. 



CoroL The force / being found, the square of the velocity may be deduced 

 from the equation v^ ■=./ X c + h, and the time from the fluent of the fluxion 



V //(/"x c + h)" 



2. Let the body move in a curve of double curvature, and let the forces /-', 

 /", &c. tending to all the points m', yi'\ &c. (except two, m, and m',) be given; 

 to find the forces tending to the points m and m'. 



