38 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



co-ordinates) (a, a', 1) are the co-ordinates of the centre, and a" the radius of the 

 circle A, then A° stands for (x — azf + (y — a'zf — a"V, and the like for 

 B°, C , D°. Write also 



a : b : c : d = BCD : - CDA : DAB : - ABC 



where BCD, &c, are the triangles formed by the points (B, C, D), &c. ; the 

 analytical expressions are 



a : b : c : d : 



h, 



v, 



1 



: — 



c, 





1 





d, 



df, 



1 





c, (', 1 





d, d', 1 



: — 



a, a, 1 



d, d', 1 





a, a, 1 





b, V, 1 



a, "', 1 





i, V, 1 





c, c', 1 



so that 



a +b + c +d = , 



a« + bb + en + dd = "t , 

 ;l ,/ + y/ + cr ' + d(/' = ; 



this being so, it is clear that we have 



aA° + 1»B° + cC 3 + dD° = 

 * 2 [a0 2 + «' 2 -a" 2 ) + b(/> 2 + ^-i" 2 ) + c^ + d 2 - O + d(d» + d'*~ (£*)] = KJ. = K , 



a constant. 



87. I am not aware that in the general case there is any convenient expres- 

 sion for this constant K ; it is = when the four circles have the same ortho- 

 tomic circle ; in fact, taking as origin the centre of the orthotomic circle, and its 

 radius to be = 1, we have 



« 2 + ft' 2 — a"- = 1, &C. , 



whence 



i$r=a + b + c + d=n; 



that is, if the circles A,B, C, Z>have the same orthotomic circle, then A", B~, O, D c , 

 a, b, c, d, signifying as above, we have 



aA° + bB° + cC° + dD° = , 



and, in particular, if the circles reduce themselves to the points A , B, C, D re- 

 spectively, then (writing as usual A, B, C, D in place of A°, B°, C°, D c ) if the 

 four points A, B, C, D are on a circle, we have 



aA + bB + cC + dD = . 

 88. This last theorem may be regarded as a particular case of the theorem 



aA + bB + cC + dD = Kz 2 = K, 



viz., the four circles reducing themselves to the points A, B,C, £>, we can find 

 for the constant K an expression which will of course vanish when the points 

 are on a circle. For this purpose, let the lines BC,AD meet in E, the lines 



