36* PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



anti-points (0,, Z) 3 ) will lie on the hyperbola. Suppose (A, B, C, D) are coney clic, 

 then, as noticed, AB and CD will be equally inclined on opposite sides to the 

 transverse axis of the ellipse — the conjugate diameters OF, OG will therefore be 

 equally inclined on opposite sides of the transverse axis— and the points F and G 

 will therefore be situate symmetrically on opposite sides of the transverse axis, 

 that is, the points F and G will respectively determine the same confocal hyper- 

 bola, and we have thus the required theorem, viz., if {A, B,C,D) are any four 

 coney clic points on an ellipse, or say on a conic, and if (A 3 , B.,) are the anti- 

 points of (A, B), and (C\, D 3 ) the anti- points of (C,D), then (A 3 ,B 3 , C 3 , D 3 ) will 

 lie on a conic confocal with the given conic. 



System of the Sixteen Points, the Axial Case — Art. Xos. 81 to 85. 



81. The theorems hold good when the four points A, B, C,D are in a line ; the 

 anti-points (B v CJ of (B, C), &c, are in this case situate symmetrically on oppo- 

 site sides of the line, so that it is evident at sight that we have (A v B v C v D r ), 

 (A, v B 2 , C. v D 2 ), (A 3 , B 3 , C 3 , D 3 ), each set in a circle ; and that the centres 

 R, S, T of these circles lie in the line. The construction for the general case 

 becomes, however, indeterminate, and must therefore be varied. If in the general 

 case we take any circle through (i>, C), and any circle through {A, D), then the 

 circle R cuts at right angles these two circles, and has, consequently, its centre 

 R in the radical axis of the two circles ; whence, when the four points are in a 

 line, taking any circle through (B, C), or in particular the circle on BC as 

 diameter, and any circle through (A,D), or in particular the circle on AD as 

 diameter, — the radical axis of these two circles intersects the line in the required 

 centre R, and the circle R is the circle with this centre cutting at right angles 

 the two circles respectively ; the circles S and T are, of course, obtained by the 

 like construction in regard to the combinations (C, A ; B, D) and {A, B j C, D, 

 respectively. It may be added, that we have 



R \ ( extremities R \ (B,C;A,D, 



S > centre and < of diameter S J-sibiconjugate points of involutions \ C, A, B, D, 

 T) (of circles T) \A,B, C,D, 



and that (as in the general case) the circles R, S, T intersect each pair of them at 

 right angles ; and they are evidently each intersected at right angles by the line 

 ABCD (or axis of the figure), which replaces the circle in the general case. 



82. If the points A, B,C,D are taken in order on the line, then the points 

 is?, S, T are all real, viz., the point R is situate, on one side or the other, outside 

 AD, but the points S and Tare each of them situate between B and C ; the 

 circles R and T are real, but the circle S has its radius a pure imaginary 

 quantity. 



83. If one of the four points, suppose D, is at infinity on the line, then the 

 anti-points of (A, D), of (B, D), and of (C, D) are each of them the two points 



