1887.] A. Mukhopadliyay — Memoir on Plane Analytic Geometry. 325 



the value of 



1 1 



Pi P2 Pz Pi 



is found to be 









r 



and the locus in 



question is 









a^^h^~ 



-e+P- 



..)■ 



Similarly, if we 



have the parabola 









y^ = 4iax, 





the value of 





1 1 



Pi Pa P3 P4 











ia 





1 





and the locus sought is 



Lastly, as in the equilateral hyperbola, we have (a-{-h) =0, the required 

 conic-locus is the given conic itself, and we have the following 



Theorem. — If through a given point P in the plane of any conic, 

 any two chords be drawn mutually at right angles, the sum of the 

 reciprocals of the rectangles under the segments is constant ; and, for 

 different values of this constant, the locus of P is a family of concentric, 

 similar and similarly situated conies, which, however, all merge into the 

 primitive conic when it is an equilateral hyperbola. (Cf. Salmon's 

 Conies, §. 181, Ex. 2, Ed. 1879, p. 175). 



§. 25. Theory of Envelopes. 



§. 25. On Three Parabolic Envelopes.— As an illustration of the 



theory of envelopes, we proceed to discuss the envelopes of the sides of 

 all equilateral triangles inscribed in a given triangle. 



Let ABC be the given triangle, and 

 A'B'C an equilateral triangle inscribed in 

 it ; let r be the side of this equilateral 



triangle, and let Z.AC'B' = --f ^, so that 

 ZA'C'B = 1-0, ZBA'C = ^-\-e - B, 



