44 Rectification of the Ellipse. 



and dz=dx sin, o (1 -a;^ gin, ^9) " ^. Developing and 



-v/=T ^r^sin.^ 3. S.x^ sin. ^9 



z=log,(t]) +^sin.9+-j-;2ry" + lT2r3:475 +^^'^- 



or z= log. (11) ' + circular 

 from which it would appear that 



Jdxs 

 We come to the same conclusion, by calculating dz directly Jroni 



the equations 



dy^±2 COS. cfdxdy+dx-=dz^. 

 From (23) and (24) we have 



x^=- ~ ^^; and y 



+ ^g-?A/-» _g9V"'\ £29^/-! _ J 



From these two equations come 



sin. cpjydxz 



^^(3?+.^V->_^-^^v/-')^3Y_{_2^,^*a/-i_2H-C 



t4ie"^^-'-l) 



and 



sin. (p/xdy 



^^(.?+3zV-'_e-^^v^->)^-j_2^e'?^-'+2z+C' 



+4(e 



9a/- 



and then —^f{xdy — ydx) and — ^ — f{ydx—xdy) the ellip- 



tic sections, bounded by an axis, the radius vector and curve. 



We arrive at the equations (1) and (2) by simply changing the 

 direction of the axes of the co-ordinates of a^y^ ^h^x^=a^h'^ 

 where x and y are supposed to be at right angles. It is evident from 

 what has been said that x=- cos. 9 (^'+2/0 ^^^ 2/=sin. 9 {x^ — 

 but dx and dy have different signs, as is seen from a^y- -{-b^x^ =:a^b 

 consequently, dx— —cos. 9 [dx^-^-dy') and dy=sui. {dx' — dy^) 

 dropping the accent, and we have, after substitution 



/a^ sin.^9— &3 cos.^ 9\ ^ a-&^ 



^ ^ ^y\a' sin.^ 9+^3 ^os.^ 9/+^*' "a^ sin.^* 9+i^ cos.^9 

 inAdy^'\'d^^=dz-—y^^2dxdy{Qos.^ 9— sin.^* 9) +dfa:3 



9 being arbitrary 



. , a* sin.^ 9 — 6= cos.^ 9 



COS.2 9 — sm.^ 9 = 17-^^ — 5 — TT^ — ^^~ 



I 



