1890.] connected with Bicircular Quarties. 89 



U r be drawn meeting the quartic again in the system of points a', 

 then the systems ft, a' will also be residual. For since the systems 

 a, ft make up the intersection of a curve U p with the quartic, and 

 a', ft' make up its intersection with a curve U r , the four systems 

 together make up the intersection with the quartic of a curve 

 whose order is 2 (p + r) ; but the systems a, ft' together make up 

 the intersection of the quartic with the curve U q of order 2q, and 

 therefore by § 4, the systems a, ft together make up the complete 

 intersection of the quartic with a curve whose order is 2 (p + r — q). 



iii. Two systems which are coresidual to the same are co- 

 residual to each other. 



If ft and ft' are coresidual as having a common residual a, and 

 if ft', ft" have a common residual a ; then by the last theorem a 

 is also a residual of ft", and a' a residual of ft ; that is, if ft, ft" are 

 each of them coresidual with ft', then ft, ft" are coresidual with 

 each other, for a, a' are each of them a residual of ft, ft". 



7. Suppose now that we have given a system of 4<p + 1 points 

 on a bicircular quartic, through them we may draw a circular 

 curve of order 2 (p + r) and we obtain a residual system of 4r — 1 

 points ; through these we may draw a circular curve of order 

 2 (r + s) and the residual system will consist of 4s + 1 points ; 

 through these we may draw a circular curve of order 2 (s + 1) and 

 we obtain a residual consisting of 4£ — 1 points. If at auy stage 

 where we have a residual of 4?i — 1 points we draw a circular curve 

 through them of order 2?i we obtain a residual of a single point, 

 and it follows from the theorems stated in § 6 that this point 

 must be the same whatever be the process of residuation. More- 

 over whatever system of points we start with, either a system of 

 4^> + 1 points or a system of 4p — 1 points, we can always by an 

 even or odd number of stages, obtain a single point which will be 

 a coresidual or a residual of the given system, according as the 

 number of points in the given system is 4>p + 1 or 4jj — 1. 



The principles just established enable us to find, by means of 

 circular constructions, the point residual or coresidual to any given 

 system of points, the number of which is 4p ± 1. 



8. To find the coresidual point of a system of five given points. 

 Let P v P 2 , P 3 , P 4 , P 5 be the points, through any three of these 



Pj, P 2 , P 3 say, draw a circle cutting the quartic in Q v and through 

 the other two P 4 , P 5 draw a circle cutting the quartic in Q 2 , Q s , 

 then the circle QiQ»Q 3 will cut the quartic in the point R which 

 will be the coresidual of the given system. 



Let the points P l5 P 2 , P 3 , P 4 , P 5 coincide, then we see that to 

 obtain the coresidual of five consecutive points P, we have to 

 draw the circle of curvature at P meeting the quartic in Q, and 

 one of the bitangent circles at P touching the quartic again at Q x , 



