VOL. LXXVIII.] PHILOSOPHICAL TKANSACTIONS. QQl 



can be deduced 60 = y : op = i :: pm = y : lm = ^; but lm + sm = ^ -}- x 

 __ yx_—^ __ j^g. gj^j consequently Yp •=. is/ {x^ -j- y^) '.po=:y :: ls = ^zl^. gy 

 = ^i^-o~ ^ rv ; but SY is a function to be deduced as above of sp = t/ (o;''^ 4- v^), 



whence the fluxional equation ^^-TV^ gs = f * (-^^ + i'^)* 



2. Fig. 10. Let a body be acted on by any number of forces (/,/,/^/", &c.) 

 in the same plane, tending to the given points s, s', s^^, s"', &c. ; to find an equa- 

 tion expressing the relation between sp = d and s'p = jy\ and their fluxions, 

 where p is a point situated in the curve which the body describes. — Suppose yp a 

 tangent to the curve at the point p, and pz perpendicular to it; and resolve all 

 the forces tending to s, s', s''', &c. respectively into two others; one in the direc- 

 tion PY, and the other in the direction pz; substitute for sp, s'p, s"p, s'"p, &c. 

 respectively d, d'', d'', h"*, &c.; and suppose sy, s'y', s"y", s'"y"', &c. perpendi- 

 cular to the line p"y : then will the triangles pqt and spy, paV and s'py' be si- 

 milar, where pa denotes a very small arc, and qt and qt' are perpendicular to 



,1 1- J ' U FT X SP D X D Pt' X s'p d' X d' , 



the Imes sp and s p ; hence pq = = = ; — = --—7- ; and 



PY PY PY PT 



consequently py : py' :: d X d : d' X d'; and if the quantities d, d*, d and d' are 

 given, the ratio of py : py' will be given; which being given, together with the 

 line ss' = a, the lines py and py', sy and S'Y', can be found; for, drawing sl 

 parallel to py, and meeting sV in l, let py' = w X py, then yy' = (wz jb ' ) 



PY = sl, sy = V/(SP^ — PY^) = >/(d^ — PY^), sV = /(S'P^ — Py'^) = (d'^ — 



w' X py"), ls' = S'Y' + SY = ± /(d' — w^py") ± -/ (d' - PY^) ; and ss'^ = 

 sl'^ -{- Ls'% an equation in which all quantities (except py) are given, and con- 

 sequently PY is determined by an equation, which will be a quadratic; but py 

 being found, from thence py', sy and s'y' may be deduced, which are conse- 

 quently all functions of d, d', d, d', and invariable quantities; and their fluxions 

 py', sy, and s'y', functions of d, d', d, d, d, and i' : from the similar triangles 

 before given Sy : pa = — :: py : -^ = po the radius of curvature; hence po is a 

 function of d, d', d, d', and i', if i = 0; and from d, d', ss', d, d', and the 

 point s'^ given in position can be determined s''p, s'-'y^'' and py''''; for let s"h =z c 

 be drawn perpendicular to ss' = a, and sk = b; then will sl (if p/ be a perpen- 

 dicular from the point p to the line ss') = + ^ — ° ~" , and s'/ = 4- "''+^^—^* 

 and p/ = /(sp^ — sP), and s'^p = ^/{{b ± slf + (c ±p/)'); draw s'V per- 

 pendicular to the tangent py, and cutting the lines ss' and sk parallel to py in 



J \' ^ 4.U -11 j: ^ X a/Tss'*— (py ± py')') „ c x ss' , 



andw respectively; then will oh= (py ± fy' )""^/ ' ^ ^ = ~T~* (^"*^ 



from the similar, triangles s"oh and son) on = {b ± oh) X -tt ; whence s''''y'' = ± 

 s^o ± on ±^SY will be a known function of d, d', jj and o, and invariable quan- 



