106 
CIRCLE. 
dam communem in Propositione descriptam or, in Playfair’s 
translation of these words, “ a Porism is a proposition, in which 
it is proposed to demonstrate, that one or more things are given, 
between which and every one of innumerable other things, not 
given but assumed according to a given law, a certain relation, 
described in the proposition, is to be shewn to take place.” 
Lhuilier: Elémens cV Analyse Géométrique et tV Analyse 
Algébrique, p. 39. 
Lamé : Examen des differentes MétJiodes employees pour 
résoudre les Problhnes de Geometrie, p. 22. 
6. To find on the circumference of a given circle a point 
such that the sum of the squares of its distances from two given 
points shall be equal to a given area. 
Let the equation to the given circle be 
x 2 + y 2 = c 2 (1) ; 
and let (a, &), («', ¿'), be the coordinates of the given points. 
Then, m 2 representing the given area, we have, by the 
condition of the problem, 
(x — a) 2 + (y — by + (x — a') 2 + (y — b') 2 = m 2 , 
or x 2 +—[a + a!) x — (b + b') y = [ni 2 —a 2 — li 2 — a' 2 — b' 2 )... (2). 
From (1) and (2) we have 
(a + a!) x + (b 4- b') y = \ (2c 2 — m 2 + d 2 + b 2 + a 2 + Z>' 2 )...(3). 
Thus the required point will be an intersection of the circle 
(1) with the chord (2). 
COR. Suppose that 
a + a! — 0, b + b' = 0, 
and 2c 2 — m 2 + a 2 + b 2 + a' 2 + b' 2 = 0, 
or d 2 + b 2 + c 2 = \ni 2 — a 2 + b' 2 + c 2 . 
Then the equation (2) becomes an identical equation, and 
the problem becomes indeterminate ; any point whatever in (1) 
satisfying the conditions of the problem; the required point 
being thus replaced by a circular locus of appropriate points.