38 



Kectification of the Ellipse. 



Art. IV. — On the Rectification 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 be 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 

 of F is an ellipse, AA'BB^ 



Put a;=:CE=DF, ?/=CD=EF, DE equal to unity, and (p for 

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

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

 CA' and CA, and that cp is estimated from the line CB'. 



Draw £?/! indefinitely near to DF, and through/ draw fe, parallel 

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



gF' -2 cos.cpgfgF+gf-^ ==Ff ; 

 and at its limit dy^ —2dxdy cos. cp-^-dx^ =dz^, z being the arc, AF. 

 Passing to the factors of the first member of 



y"^ —2xy cos. (p+a;^ = l, and we have necessarily 



+zi 



?/ — a?(cos. (p — sin. (pi/ — i)=e and 



-z' 



y — a?(cos. 9+ sin. (p-v/ — l)=6"'" ' 

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



to be determined. 



+z' 



Differentiating and dy — dx{cos. cp — sin. (p-v/ — i) =^tdz'e~ ' 



dy — dx[cos. (p-\- sin. (p-v/ — i) =^^dz'e'^ ? 

 multiplying and dy~ ^2dxdy cos. (p-\-dx^ =■ — dz'^ ; 

 integrating we have f'^dy'^ —2dxdy cos« (p-^dx^ =s"v — 1 z=z. 



