The American journal of science and arts . - e ^ ) commences at A, and terminates at A, com-prising any number of circuits together with AA% log. (+e ^ ) * comprises any number of circuits, commenc-ing at A and terminating at B, counting positively, log. (4 e ) commences at A and terminates at Band log. ( —e^^ ) begins at A, and ends at A, counting nega- tively. When (p is a right angle w2^=log. \:^\/^) When tx=y it indicates the vertex of the greater and lesser axis,as is seen by making the first member of y^ -\-2xy cos. 9+a?^ =^CF^maximum; writing therefore la; for y, in (15) and (16) and we
The American journal of science and arts . - e ^ ) commences at A, and terminates at A, com-prising any number of circuits together with AA% log. (+e ^ ) * comprises any number of circuits, commenc-ing at A and terminating at B, counting positively, log. (4 e ) commences at A and terminates at Band log. ( —e^^ ) begins at A, and ends at A, counting nega- tively. When (p is a right angle w2^=log. \:^\/^) When tx=y it indicates the vertex of the greater and lesser axis,as is seen by making the first member of y^ -\-2xy cos. 9+a?^ =^CF^maximum; writing therefore la; for y, in (15) and (16) and we havenz=log. a:(l-e^)^=AG; (17) -w^=log. a;(l-e^^^)^~ = -AG; - (18) Rectification of the Ellipse. ?nz- then (17) - (IS) nz=log. and(19)-(20) -n^=log.( ^7^1^) =AG 1-e = -AK 41 (19)(20) (21)(22) consequently, (21) - (22) nz = \og. {±1^ )=AG+AK = GKequal to the elliptic quadrant. Drawing the figure below, and following the description at thecommencement, it is evident that we have CD^+2CE CD +CE2=CF2 or 2/2+2c-c?/COS. cp+x^ = Parting the first member of this equation into its simple factors, +2; we have y-i-x (cos. 9 —sin. (pi/_i) = e and —z y-{-x (cos. (p + sin. (p\/JZ^)=e+ ; differentiating both of these equations, observing from xdx sin.^ cp dy=- —dx cos. (p^ , :::==? that dx and dy have different signs, V 1 — a?^ sin.^ (p +z we have dy — dx (cos. 9— sin. cpv/ —i)=l<^^e and —z dy — dx (cos. (p+sin. (py — 1) ^l^dze^multiplying, dy^—2dxdy cos. (p-\-dx^ = — dz^ the differentialrelation of the circular arc AF, and its co-ordinates, as is seen by Vol. XVIII.—No. 1. 42 Rectification of the Ellipse. the figure, hence, what has been said of the ellipse, ABBA^ is equal-ly applicable to the circle ABBA^ Developing (15) and (16,) y being treated as the greater, we have wzv/in=log.(y)-^e-?-/^-2pe-^9v^=;^-3^3e-=^9v/--_—g-4(pv-i _ etc. and -*i^^/^=log. (2/)-^e<Pv^-|p==<Pv/^ - ^3e3?v/^_ X* .— —^g4(pv/-i — etc. then taking the difference
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Keywords: ., bo, bookcentury1800, bookpublishernewhavensconverse, bookyear1820
