518 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Hence, if 
\jj(xyzw)-^a hl x 2 -\-2a ltZ xy-\- . . . ; 
the distance 8 between the points is given by, 
— B~.K = a hl (x—x')~ + 2a h , 2 (x—x')(y—if)+ . . . 
=\p{x—x', y—y, z—z', w—w'}. 
Thus, if Bs be a small element of arc, we shall have 
— K.(Bs)~=ip{Bx, By, Bz, Bio }. 
(88) 
(89) 
61. If P, Q, It be any three points, then by equation (15) we have for the area of 
the triangle 
-16{A(P,Q.B)} , =n(J^» 
and by § 8 
n 
0, P, Q, E 
e. P, Q, E 
Jxn 
1. 2, 3, 4 
1, 2, 3, 4 
= n 
0, P, Q, E 
1, 2, 3, 4 
Hence if (aqy^p^q), (x. 2 yoZ 9 W-f (x 3 y 3 z 3 w s ) be the coordinates of the points P, Q, R 
referred to any system of circles (1, 2, 3, 4) we shall have 
A(P, Q,R)=p Xl ’ Xs ’ kl ; .( 90 ) 
where, 
-4AVW.H 
ay, 
x 2 , 
co 
Vi> 
y 
y 3 > 
2 i> 
Zo, 
y\, 
w s , 
'1, 2, 
3, 4\ 
T-n( 
f 2, 
3.4 = 
Coordinate Systems of Reference, —§§ 62-66. 
62. There are two systems of circles which are convenient as systems of reference— 
(i.) a system consisting of four circles cutting one another orthogonally,* (ii.) a system 
of two circles cutting orthogonally, and their two points of intersection. The former 
has been called the “ orthogonal ” system, and was first used by Darboux, ‘ Sur une 
Classe remarquable de Courbes et de Surfaces algebriques ’ (Note X., 1873). The 
latter system might be called the “ semi-orthogonal ” system; it is mentioned by 
Mr. Homersham Cox in the paper “ On Systems of Circles and Bicircular Quartics ” 
(‘Quart. Journ. Math.,’ vol. 19, 1883, p. 116). 
* [Casey uses five orthogonal spheres—“ Cyclicles ” (1871), p. 600. But the Erst use of four mutually 
orthotomic circles was, I believe, by Clifford in a series of questions proposed by him in the * Educa¬ 
tional Times’ for 1865-6. See Reprint, vol. 6.—October, 1886.] 
