240 Journal of the Asiatic Society of Bengal. [April, 1908. 
So that, if the locus of 2’, y’ is the curve f(z, y)=0, the locus of 
A, ¥ ia f (2 z.3) = 0, which agrees with Dr. Mukho- 
ate at ps 
padhyaya’s result and which at the same time gives a similar 
result for the hyperbola. 
t is wre to observe that the relation between the 
two points a’, y’ and X, Y isa arias “se one, so that each is the 
middle point of the polar chord of other, as is evident from 
the expressions for the co-ordinates of ‘ee one in terms of those of 
the other—the relation being, in fact, exactly similar to that of 
two inverse points. f(z, y)=0O and f tue ae a a e =0, are, 
na a8 git Ba 
ee two inverse loci, each ee the locus of the middie 
ts of polar chords of points lying on the other locus 
4, Similar Danial a also true for the parabola es, 4ax. 
If #, y be the co-ordinates of any point P and X, Y, those of the 
e point of the ae dient of P with respect ‘to the parabola, 
it has been proved by Dr. Mukhopadhyaya 
. y® — 2ax 
y* =2az + 2aX 
or Y?=2aa+ 2aX 
Y*?—2aX 
cage ee as 
so that the points 2, y Pong X, Y may, in a certain sense, be 
— as inverse point: 
the case of the Raoatoi me above results can be estab- 
lished. directly with very great e 
Thus if (a, y) : (X, Yb os ie acuctianstie of the points P, 
in the opposite figure, since PQ is parallel to the axis (Prop. XX 
p. 38, Dr. Mukhopadhyaya’s ater of Conies). 
Y= y. 
Again, if R be the middle point of PQ, F& is a point on the para- 

: ; a+ X 
bola and its co-ordinates are zi a Yy, 


a+ X 
2 
er ( ) =2a2 +2aX whence X= ¥ =, 
