484 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
of the parallelogram formed by joining the points, in which two parallel tangents to a 
hyperbola meet the asymptotes, was called the “ Power of the Hyperbola”—and the 
name was borrowed by Steiner, in the paper quoted above, which was written in 1826, 
to express the constant rectangle of the segments of any chord of a circle through a 
point, and this rectangle he called the power of the point with respect to the circle. 
Steiner also extended his definition thus : if O be one of the centres of similitude of 
two circles whose centres are A, B : and if a chord through 0 cut the circles in P and 
Q respectively, but so that AP, BQ are not parallel, then he proposed to call the 
rectangle OP, OQ the power of the two circles with respect to 0. 
Darboux seems to have been the first to give the definition of the power of two 
circles, as used in this memoir, in a paper written in 1872 and published in the 
‘Annales de l’Ecole Normale Superieure/ vol. 1 . Clifford also gives the same 
definition in a paper, the probable date of which is said to be 1866, given in the 
Appendix to his ‘Mathematical Papers’ (1882); but the paper itself does not seem 
to have ever been published. 
Mr. Homersham Cox, in a paper published in the ‘Quarterly Journal of Mathe¬ 
matics ’ (vol. 19, 1883), has shown that the power of two circles may also be defined 
as the product of two circles. 
3. If the equations of two circles be 
x 2 -\-if+2g 1 x+2f l y+c 1 =:0, .(l) 
x 2 +y 2 +2g 2 x +2/ 2 y+c 3 = 0,.(2) 
we have 
^i,3=c 1 +Co-2p 1 po~2/ 1 / 3 .(3) 
4. It will be convenient to define the power of a straight line and a circle as twice 
the perpendicular from the centre of the circle on the line; and the power of two 
straight lines as twice the cosine of the angle between tliem. # 
Thus the power of (1) and the straight line 
is given by 
— 2x cos a — 2y sin a fi-2p=0,.(4) 
7 r = 2 p -f 2 g x cos a+ 2/j sin a.(5) 
Thus regarding (4) as a degenerate form of (2), (5) may be considered as a particular 
case of (3). 
[* We can easily stow that these definitions are included in tliat given in § 1. Thus, considering a 
straight line as a circle of infinite radius, B say, the power of a circle, radius r, with respect to it 
= (p J r B) 2 —r 2 — B 2 =2pB, in the limit, p being the perpendicular from the centre of the circle on the 
straight line. Similarly the power of two straight lines, inclined at an angle tc, =2I2 3 cosw. Conse¬ 
quently, as we are going to deduce our results from a certain symmetrical determinant, we may ignore 
these factors B, B 2 , and define these powers as in § 4.—22nd October, 1886.] 
