Rectification of the J^Utpsc. 



41 





X 



-9\/-') 



x/- 



I 



AK 



x{\ 



e 



(pv/-iN-v/- 



AK 



Uien (17) - (18) nz 



log 



1-e 



^ipv/- 



1-e 



(p\/- 



v/- 



(i9) 

 (20) 



I 



AG 



(21) 



and (19) -(20) -nz=log 



1 



e 



-!Pv/- 



1 



.(t 



v/- 



I 



AK 



(22) 



v/ 



AG+AK-GK 



consequently, (21) -(22) ?iz=log. 

 equal to the elliptic quadrant. 



Drawing the figure below, and following the description at the 

 commencement, it is evident that we have CD^-f-2CE CD cos. 

 ACB'+CE3=CF^ ory^+2:r^cos. ^-\-x^ = \. 



Parting the first member of this equation Into its simple factors, 

 we have y-\-oc (cos. 9 — sin. 9\/ 



tz 



and 



y-\'X (cos. (p-f-sin. 9\/ 



-z 



e 



+ 



differentiating both of these equations, observing from 



dy 



dx cos. tp" - — ^^= 



Vl—x 



xdx sin-^ (p 



sin.^ (p 



that dx and dy have different signs, 



+z 



we have dy — dx (cos. 9 -sin* (p\/ — i)=j:£?^e and 



dy — dx (cos- cp+sin* (p^/ZTi) =^£?ze^ 

 muluplying, dy- —2dxdy cos. tp-f-J^a — _j^2 the differential 

 relation of the circular arc AF, and its co-ordinates, as is seen by 



Vol. XVIIL—No. 1. 



6 



