. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature. Encyclopedias and dictionaries. equation (the fluxion of y being negative), pyy â f zay-â _ a ⢠J- ⢠2 a j' - j--"\ â 'â (by writing for _y its equal), ; f Now, by fubftituting â-ââ 2 12 that "^^ for v% and â for y, f â¢we have ' + . 2 a \ B« = x1, and n b = y, when, by 47 E. 1 B b -s *' V ^* . â will be = x'''+' »i~*\iJ *' " + y' = B F or B ;⢠; and B b I b. n y y' + y rs\>, y +/" : v :â-= -, , x'' \ y \ri : : B C : C F, that is, \ = C F or C r ; the in- crement of which is rfj that i
. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature. Encyclopedias and dictionaries. equation (the fluxion of y being negative), pyy â f zay-â _ a ⢠J- ⢠2 a j' - j--"\ â 'â (by writing for _y its equal), ; f Now, by fubftituting â-ââ 2 12 that "^^ for v% and â for y, f â¢we have ' + . 2 a \ B« = x1, and n b = y, when, by 47 E. 1 B b -s *' V ^* . â will be = x'''+' »i~*\iJ *' " + y' = B F or B ;⢠; and B b I b. n y y' + y rs\>, y +/" : v :â-= -, , x'' \ y \ri : : B C : C F, that is, \ = C F or C r ; the in- crement of which is rfj that is, fuppofmg x' to be inva- y" -i-yy" x x7* +/â â ] i - y'y'xjy'^ riable) , *ZZZHi__ _ 2y" +/* + y«"y" ^TT - rf. Again, the triangles E £ b and E r / being fimilar, B b - r f: R b â ⢠(BE - r E, or) r B : B E ; that is, («f« +.y/)| _. neral expreffion for the radius of curvature of all curves referred to a fixed or central point, when *,' or x is inva- riable. Hence, if .v be made =r 1, the general expreffion for the Wliere- radius of curvature will be -^^ I ^ ^_] fore, if we put the equation of the given fpiral into fluxions, making ^? = ,, and put this fluxional equation into fluxions again; and from thence, or from the na- ture of the curve, find the values of i" and y : then, if for;- and y we fubftitute thefe their values in this ffene- ralexprelhon, we ihail have B E the radius tif curvature re- quired. Otherwife. Let A R B he the propofedcui-veyiftfe - ) P the point, or centre, to which its ordinates are referred. JN O L the evolute, and R O the ray of curvature at R Moreover let P H be perpendicular 'to R O ; an,', fuppofl jng the ordinate PR. (y) to I )tion of the point K alon fA Rand P H (/.), exprefling the eel . , jn1 , ;>_ aâd H '? dlre«K" '' n ir to R O, be denoted by i and p refpeftiyely : the flu do s if quantitii 1 b the celerities by wl I in magnitude. (S;, ion), Therel re 1 celer
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Keywords: ., bookcentury1800, bookdecade1810, booksubjectencyclo, bookyear1819
