494 
MR R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Hence by (20) 
774,4. i n 
1 , 2 , 3 
1, 2, 3 
-•~..Ki3)WK*3}W{<n 
But if r be the radius of the common orthogonal circle, we have by (16), 
1, 2, 3' 
o_ n \l, 2, 3/ 
n 
1,2,4\ 
1 ’ 2 ’ 4 ' &c. 
Therefore 
32{A(1, 2, 3)} 2 32 |A(1, 2, 4)} 
tt.v, 4-A(l, 2, 3) = 7T.,. ].A(2, 3, 4) +7T.,, S .A(1, 4, 3)+7r. r> 3 .A(1, 2, 4). 
( 21 ) 
Thus if any four circles have a common orthogonal circle, and the areal coordinates 
of the centre of one of them referred to the triangle formed by joining the centres of 
the other three be a, ft, y : then the powers of any other circle with respect to these 
four circles are connected by the relation 
7T iTj — a • v j 7 ^ ] + ft. TT Xi o + y.7r. r) o 
( 22 ) 
As a particular case we obtain the well-known theorem that if A, B, C be the 
centres of any three circles, P any point on the circle which cuts these orthogonally, 
and 0 any other point, then 
OP 2 = (0 A?-r*)a + (OB 3 — ;y)/3+ (OC~ — r*)y ; 
where a, /3 , y are the areal coordinates of P referred to the triangle ABC. 
Orthogonal Systems. —■§§ 25-29. 
25. Four circles may be said to form an orthogonal system if they cut one another 
orthogonally: it is clear that the centre of any one of them is the orthocentre of the 
triangle formed by the other three. 
If the system be denoted by (1, 2, 3, 4), then ( x , y) being any other circles, the 
equation 
n 
becomes 
b 2, 3, 4 
y, 1,2, 3, 4 
= 0 
y> 
77 ./', 1 ? 
7 T?^ 
77.1-, 3, 
77 -/', 4 
77 V, 1 ’ 
77 i,n 
0, 
0, 
0 
77 '/, 2 > 
0, 
o 5 
0 , 
0 
77 y, 3 ’ 
0, 
0 . 
77 ;,. 0 , 
0 
7 T !/A> 
0, 
0 , 
0 , 
774,4 
