PROBLEMS AND SOLUTIONS. 
605 
[485 
485] PROBLEMS AND SOLUTIONS. 605 
1, and that 
and, similarly, 
bs at these 
id ~ yo) (xx x + yyi -1 ) + (y~ 2/i) (xx 0 + yy 0 - 1) = (ax + fiy + y) (7y + £)> 
r shortness 
(y ~ yo) (xx x + yyi - 1 )~{y~ 2/i) (xx 0 + yy 0 - 1) = fixy - ay 2 + yx + a. 
11. The equation in question may be written a 2 P + b 2 Q = 0, where 
•rdinates of 
p = (x - x 0 ) 2 (xx x + yy 1 - l) 2 -{x- x x ) 2 (xx 0 + yy 0 - l) 2 , 
rm to this 
Q ={y- yof {xx 1 + yy x - l) 2 - {y - y,y (xx 0 + yy 0 - l) 2 , 
. Or what 
values which are given by means of the formulae just obtained; there is a common 
it is to be 
factor ax + f3y + y which is to be thrown out; and we have also, as is at once 
hen restore 
verified, ~-- ^ ^ , so that these equal factors may be thrown out. We thus 
\ for 
a ab 
obtain the cubic equation 
' X oVl X ll/o- 
a different 
a 2 {<yx + a.) (fix 2 — axy — yy — fi) + b 2 (yy + fi) (fixy — ay 2 + <yx + a) = 0. 
CL 3 
This is simplified by writing x— for x, y for y. It thus becomes 
1 that the 
a 2 x [(y® - a) (fix - ay) - y 2 y] + b 2 y [(yy - fi) (fix - ay) + y 2 ®] = 0; 
transformed 
or, what is the same thing, 
a 2 x [y® (fix — ay) — afix + (a 2 — y 2 ) y] + b‘ 2 y [y y (fix — ay) — (fi 2 — y 2 ) x + a fit/] = 0 ; 
that is 
y (a 2 x 2 + b 2 y 2 ) (fix — ay) + a? [— afix 2 + (a 2 — y 2 ) xy] + If [— (fi 2 — y 2 ) xy + afiy 2 ] = 0. 
), but) for 
1 r\ T) 1 • ^ ^1 r* 1 V y§ V\ r> . . 
12. Restoring -, -, - for x, x 0 , x 1} and for y, y 0 , y x \ writing 
e line AB 
a fi y 
consequently ^^ in place of a, fi, y, if a, fi, y are still used to denote a = y 0 — y 1 , 
y, X 0 , &c. 
fi = x x —x0, y = x 0 y x — x x y 0 , the equation becomes 
y (x 2 + y 2 ) [b 2 fix — ofay] + a? [— b 2 afix 2 + (a?a 2 — y 2 ) xy] + b 2 [— (b 2 fi 2 — y 2 ) xy + afafiy 2 ] = 0, 
line L = 0, 
where now, as originally, = 1, % + = viz. this is the equation, referred to 
Qj D* 1 Qj 0~“ 
the point T as origin, of the locus of the point P considered as the intersection 
of tangents from A, B to the variable confocal conic; and it is consequently the 
equation which would be obtained as indicated in (8). The locus is thus a circular 
|uation of 
virtue of 
cubic; the equation is identical in form with that obtained (2) for the locus of the 
point at which A, T and B, T subtend equal angles, and the complete identification 
of the two equations may be effected without difficulty. 
*)> 
13. I may remark that M. Chasles has given (Comptes Rendus, tom. 58, February, 
13. I may remark that M. Chasles has given (Comptes Rendus, tom. 58, February, 
1864) the theorem that the locus of the intersections of the tangents drawn from a 
fixed conic to the conics of a system (/a, v) is a curve of the order 3z/. The confocal