28 Dr. Booth on the Rectification and 



the asymptots between the centre and the curve are, the one 

 a fourth proportional to the perpendicular and the semiaxes, 

 while the other is equal to the perpendicular itself. 



XXI. To find when the difference of the elliptic arcs is a 



maximum; m this case « is a maximum, or -j- = 0, but 



CL A 



M = -f^ ; hence 

 dK 



^ « d^ . , 



-r\ — 0, or -=—2 a VI — e^ sin^ A = 0. 



From this equation we find 



tan2x = -|- (41.) 



Deducing from this value of tan X the values of sin X, cos A, and 



substituting in (36.) and (37.), we find 



A=a{a-b\ B=b{a-b)y M-a{a-b), B=b{a-b), . (42.) 



or A = A', B = B'. 



In this case, then, when the difference of the elliptic arcs is a 



maximum, the two confocal hyperbolas become identical, and 



therefore the two elliptic arcs constitute the quadrant; this is 



the well-known theorem of Fagnani. 



To find the corresponding value of u, as 



au=: AA' =i a{a — b)f u= a — b; . . (43.) 

 also as r' = r", and r" = p, p^ = ab; . . . (44.) 



hence the semidiameter of the ellipse along the asymptot is 

 equal to the perpendicular from the centre. In this case the 

 whole quadrant is divided into two arcs whose difference is 

 equal to the difference of the semiaxes, and this point may, 

 for the sake of distinction, be called the point of linear section. 



The locus of this point for a series of confocal ellipses may 

 be shown to be the curve whose equation is 

 a^e^ = {x^ -yi) [xi + yif. 



Let tangents be drawn to the ellipse at the point of linear 

 section and produced to meet the adjacent axes; calling the 

 segment of the tangent terminated in the minor axe ^, the other 

 terminated in the major axe t\ it can be easily shown that 



/ = tan A \/a^ cos^ X + b^ sin^ A, 

 b^ tan A 



t' = 



V a^ cos^ X + b^ sm^ K 



a e^ sin A cos A 

 \/a^ cos^ A + 6^ sin^ A 

 XXII. It would be easy to show, were it not too wide a 



1 ^ j» O'C^ sin A cos A / . > * \ 7 • 



hence t — v =. — ,_ -;-- = w. . . (44*.) bis. 



'/a2cos2A + ^^sin^A 



