82 



let also a" B" A" be another ellipse whose semiaxes (A" C" and B" C") 

 are equal to a¥ and FA, and take in its circumference a point L so 

 that the semidiameter conjugate to that passing through L may be 

 equal to FP or F'F; then will the excess of the elliptic arc AT above 

 its tangent TP be greater than the excess of the hyperbolic arc A'P 

 above its tangent T'P', by twice the elliptic arc A"L. 



f'^W c' jL' 



7»MA? 



Take the point L so that when the ordinate MLH is drawn to 

 meet in H the semicircle described on the axis, the angle HC'M may 

 be equal to half the angle PCF (or P'C'F', for the triangles PCF and 

 P'C'F' have all their sides and angles equal) ; draw C"D perpendicular 

 to C"H and DE to A"a" meeting the ellipse in E ; then by lem. 2, 

 C"E will be conjugate to C'L, and by lem. 4, it will be equal to FP, 

 since the angle B"C"D is equal to HC'M and is therefore half of PCF, 

 whilst the semiaxes C"A!', C"B", are the sum and difference of PC and 

 CF. Hence the point L thus found is that required by the enuncia- 

 tion. Take p, p', h, indefinitely near to P, P', H, and similarly related 

 to each other ; let Fp and Fp' intersect TP and T'P' in q and q', and 

 with the centre C' and a radius equal to C"E describe the evanescent 

 arc Kk. Then FP, F'P' are always perpendidular to TP, TP', and 

 therefore Vq is ultimately the increment of the difference between 

 the arc AT and the tangent TP (/em. 1.) and F'q' the increment 

 of the difference between the arc A'T' and the tangent TP' ; also by 

 lem. 3. KA; is ultimately equal to LI the increment of the arc A"L. 

 Now the angles CFP and CFp are equal to CT'F and C'p'F' ;, there- 



