344 A. Mukhopaclhyay — Memoir on Plane Analytic Geometry. [No. 3, 



II. To investigate the locus of points on a system of confocal 

 ellipses, where the eccentric angle has a constant value. 

 Let any one of the confocal system be 



2 a 



where A* = a^+^^j B*^ =zh^-]-\^ ; then, if <p he the eccentric angle at any 

 point ($, >;), we have 



i' = A2 cos' </> = (aH^') cos* t 



yf = B' sinS </>=(&' + X«) sin* <p, 

 so that the locus in question is the hyperbola 



-^--r^ = ^^-&^ = c*, (178) 



cos* 9 sm^ <p ^ 



and this is evidently a member of the confocal family ; hence it follows 

 that, given a system of confocal ellipses, the locus of points where the 

 eccentric angle has a constant value is one of the confocal hyperbolas 

 which intersect the system orthogonally ; in other words, given a confocal 

 system of ellipses and hyperbolas, each hyperbola intersects the ellipses 

 at points where the eccentric angle has a constant value, and, by varia- 

 tion of this constant value, we get all the hyperbolas of the system, and 

 from a known theorem, the envelope of all these hyperbolas is an 

 imaginary quadrilateral. 



Similarly, if we have the hyperbola 



a*+A* 6HX' ' 

 which is one of a confocal system, and ^ the eccentric angle at any point 

 (1^, 1])^ we have 



^2=(a-^+X2) sec*<^, 



>;=^=(6* + A*) tan2(/>, 

 so that, if the eccentric angle has a constant value, the locus is 



-^--^=:a^-6' = c* (179) 



sec*</> tan-^"^ ^ 



and the envelope of this, for different values of the eccentric angle, is 

 the parallelogram formed by the four lines 



(cH2/'-^^)' = 4cy, (180) 



viz.^ the four lines are 



-c+2/ + ^ = 0, c-y-\-x = 0, c + y-xzzO, c + 2/ + ^=0. 

 §§. 30 — 31. Theorems on the Parabola. 

 §.30. A Dynamical Problem.— Take the parabola 

 7/* = 4iax, 

 which, when the origin is removed to a point on the principal axis at a 

 distance na from the vertex, becomes 



/ = 4a (x-j-na). (181) 



