44 Rectification of the Ellipse. 



- i 



and dz=dx sin. cp (1 —x^ sin. ^9) '^. Developing and 

 •v/— T :c='sin.='9 3. 3. x^ sin. ^9 



z=\o^.{tl) +a;sin. (p+-j^ 2. 3 +1. 2. 3. 4. 5""^®^^' 



or ^= log. (tl) * + circular arc AF 

 from which it would appear that 



Jdx sin. (p [l—x^ sin. ^(p) ^ is either an elliptic or circular arc. 



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



the equations 



y^t 2 COS. cpa:'y-{-x^=^l 



dy^t2 COS. (pdxdy-\-dx^ =dz^ , 



From (23) and (24) we have 



J^^ -^v^^ (29+^) \/^ -''v^^ 



x= : ; and y=- 



From these two equations come 



sin 



+4(e^?^-'_l) 



and 



+4(e"'^"-l) 



sin. 9 sin. 9 



and then ^ f\^dy — t/Jz) 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^ -]-b"x^=^a-b^ 

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

 what has been said that x= cos. 9 {x'-}-y') and y=sm. 9 {x' -y') 

 but £?x and f?2/ have different signs, as is seen from a~y- -{-h^x^ =a-h^ ; 

 consequently, £?:c=— cos. 9 [dx' -{- dy') diwA dy—sm. [dx' — dy') 

 dropping the accent, and we have, after substitution 



/a^ sin.29— 52 cos.^ 9\ a"h'^ 



y"^ — 22:w —r-- — ; — rrz r~ +'^^ =~r~- — : — nr^ ; — 



•^ '^ \a- sm.2 9+0^ cos.-" 9/ ' a^ sm.- 9-J-0- cos.^ 9 



and dy^ -{-dx^ =dz'^ =y^ -\~2dxdy (cos.^ (p— sin.^ 9) -\-dx^ 



9 being arbitrary, we have the right to make 



a^ sin. 2 (p — b^ cos.^ 9 



cos.^ 9 — sin. 2 9=-——: — - — —J- — 



^ ^ a^ sm.^ 9-J-62 cos.^ 9 



