38 Kectification of the JElUpse. 



Art. IV. — On the Rectijication of the Ellipse; by C. Wilder. 



Draw the indefinite right lines, AB and A'B^, intersecting in C, 

 and between the lines AB and A^B' place the given right line DE, 

 so that the point D may he in AB and the point E in A^B'; through 

 D and E draw DF and EF, parallel to A^B^ and AB, then the locus 



AA 



K. 



X 



M" 



Put x=CE=DF, y=CD=EF, DE equal to unity, and <p for 



the obtuse angle ACB', then by trigonometry y^ — 2 cos. (focy-{'Oc'^ = 1 ; 

 from this equation it is seen that x and y are positive in the direction 

 CA^ and CA, and that 9 is estimated from the line CB^ 



Draw df indefinitely near to DFj and through/ draw /e, parallel 

 to FE, then without more words it is evident that we have 



^ gT' -2cos.cfgfgF+gf^=Ff ; 



and at its limit dy^ --2dxdy cos. 9+^^:2 =£?z3, z being the arc, AF» 

 Passing to the factors of tlie first member of 



y^ —%xy cos. 9-f ^* = l5 ^^^ we have necessarily 



y — a:(cos, 9 — sin. 9 v/ — 1) =e"" and 

 y-^(cos.9-f sin. 9v/-™ i)=e"^ J 



e being the base of the Naperian system, and z' a function of x and y 

 to be determined. 



Differentiating and dy — dx(coB. 9 — sin. 9^/ — 1) ^=tdz^e' 9 



rfy — c?x(cos. 94-sin.9'v/— \)=^^dz^e^ J 



multiplying and dy^ --2dxdy cos. (p-^-dx^ = -- dz^- ; 



integrating we have f^^dy'^ —2dxdy cos. (p-^-dx"^ =x:'\/ — 1 =r. 



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