MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
483 
Casey. 
“ On Bicircular Quartics ” (1867), ‘ Irish Acad. Trans.,’ vol. 24. 
“ On Cyclides and Spliero-Quartics ” (1871), ‘ Phil. Trans.,’vol. 161, pp. 585-721. 
Clifford.* 
“ On the Powers of Spheres” (1868), £ Mathematical Papers.’ 1882. Pp. 332-336. 
“Of Power-Coordinates in general” (1866), ‘Mathematical Papers.’ 1882. 
(Appendix.) Pp. 546-555. 
Cox, H. “ On Systems of Circles and Bicircular Quartics,” ‘ Quart. Journ. Math.,’ 
vol. 19, 1883, pp. 74-124. 
Darboux. 
“ Sur les Relations entre les groupes cle Points, de Cercles et de Spheres dans le 
plan et dans l’espace,” ‘ Annales de l’Ecole Normale Superieure,’ vol. 1, 1872, 
pp. 323-392. 
‘ Sur une Classe remarquable de Courbes et de Surfaces Algebriques.’ Paris, 1873. 
Salmon. 
‘ Higher Plane Curves.’ 3rd edition, 1879, pp. 240-253. 
‘ Geometry of Three Dimensions.’ 4th edition, 1882, pp. 527-536. 
PART I.—SYSTEMS OF CIRCLES IN ONE PLANE. 
Chapter I. —General Systems of Circles. 
Definitions. §§ 1-5. 
1. The power of two circles (or of one circle with respect to the other) is the square 
of the distance between the centres of the circles, less the sum of the squares of their 
radii. 
Thus denoting the power of two circles whose radii are r l5 r 2 by tt 13 : if d h2 be the 
distance between their centres, we have 
TT - ^ ^ 
"1,2—1,2 ' 1 ' 2 ? 
or if w i 2 be the angle at which the circles intersect we have 
COS 
2. If one of the circles reduces to a point, then the power becomes equal to the square 
of the tangent from the point to the circle. In this case the definition agrees with 
Steiner’s definition of the power of a point with respect to a circle (‘ Crelle, Journ. 
Math.,’ vol. 1, 1826, p. 164). The use of the word power is of great antiquity—the area 
[* The probable date of these papers is given as 1866 and 1868 respectively. Gf. Preface to ‘ Math. 
Papers,’ pp. xxi, xxii; and also note on page 332. —October, 1886.] 
3 q 2 
