PROFESSOR CAYLEY ON POLYZOMAL CURVES. 29 



Part II. (Nos. 57 to 104). — Subsidiary Investigations. 

 Preliminary Remarks — Art. Nos. 57 and 58. 



57. We have just been led to consider the conies which pass through two 

 given points. There is no real loss of generality in taking these to be the circular 

 points at infinity, or say the points I, J— viz., every theorem which in anywise 

 explicitly or implicitly relates to these two points, may, without the necessity of 

 any change in the statement thereof, be understood as a theorem relating instead 

 to any two points P, Q. I call to mind that a circle is a conic passing through 

 the two points 7, J, and that lines at right angles to each other are lines har- 

 monically related to the pair of lines from their intersection to the points 7", J 

 respectively, so that when (I, J) are replaced by any two given points whatever, 

 the expression a circle must be understood to mean a conic passing through the 

 two given points ; and in speaking of lines at right angles to each other, it must 

 be understood that we mean lines harmonically related to the pair of lines from 

 their intersection to the two given points respectively. For instance, the theorem 

 that the Jacobian of any three circles is their orthotomic circle, will mean that 

 the Jacobian of any three conies which each of them passes through the two given 

 points is the orthotomic conic through the same two points, that is, the conic such 

 that at each of its intersections with any one of the three conies, the two tangents 

 are harmonically related to the pair of lines from this intersection to the two 

 given points respectively. Such extended interpretation of any theorem is appli- 

 cable even to the theorems which involve distances or angles— viz., the terms 

 "distance" and "angle" have a determinate signification when interpreted in 

 reference (not to the circular points at infinity, but instead thereof) to any two 

 given points whatever (see as to this my " Sixth Memoir on Quantics," Nos. 

 220, et seq.*) And this being so, the theorem can, without change in the 

 statement thereof, be understood as referring to the two given points. 



58. I say then that any theorem (referring explicitly or implicitly) to the cir- 

 cular points at infinity I, J, may be understood as a theorem referring instead 

 to any two given points. We might of course give the theorems in the first 

 instance in terms explicitly referring to the two given points — (viz., instead of a 

 circle, speak of a conic through the two given points, and so in other instances) ; 

 but, as just explained, this is not really more general, and the theorems would be 

 given in a less concise and familiar form. It would not, on the face of the inves- 

 tigations, be apparent that in treating of the polyzomal curves 



Jl(@ + L$) + Jm(& + M$) + &c. = 0, 



(O = a conic, $ = a line, as above), that we were really treating of the 



* Phil. Transactions, vol. cxlix. (1859), pp. 61-90. See p. 86. 

 VOL. XXV. PAET I. H 



