88 Mr Lachlan, On some theorems [May 26, 



hence the remaining (p — l) 2 points must lie on the curve of the 

 2pth order. Hence the truth of the theorem enunciated is 

 manifest. 



3. An important particular case may be thus stated : Every 

 circular curve of the 2nth order which passes through 4?i — 1 fixed 

 points on a bicircular quartic must pass through one other fixed 

 point*. r 



4. From this theorem we may deduce the following : If of 

 the 4<(m + n) intersections of a circular curve of the 2(m + ?i)th 

 order with a bicircular quartic, 4>m lie on a circular curve of the 

 2mth order, the remaining 4m lie on a curve of the 2/ith order. 



For let U m denote the curve of the 2mth order, and let a curve 

 U of the 2?ith order be described passing through 4m — 1 of 

 the remaining 4n points ; then these curves U m , U n together make 

 up a circular system of the 2(m + ?i)th order passing through 

 4 (m + n) — 1 points on a bicircular quartic, this curve must there- 

 fore pass through one other fixed point, which must clearly lie on 

 TJ n ; also this point must be a point in which the given curve of 

 the 2 (m + ?i)th order meets the bicircular quartic ; hence we see 

 that the theorem stated above must be true. 



5. If now we have two systems of points a, ft, which together 

 make up the complete intersection of a bicircular quartic with 

 a circular curve of any degree, i.e. if a + ft is a multiple of 4, one 

 of these systems may be called the residual of the other. Since 

 through a given system of points, any number of curves of different 

 orders may be described, it is evident that a given system of points 

 a has an infinite number of residual systems ft, ft', &c. Two 

 systems of points ft, ft', may be called coresidual systems if both 

 are residuals of the same system a. 



6. We have at once the following theorems : 



i. Two points which are coresidual must coincide. This is 

 merely a restatement of the theorem in § 3 ; for if through 4n — 1 

 points a on a bicircular quartic we describe two circular curves 

 of the 2 nth order, meeting the quartic again in the points ft, ft', 

 then ft and ft' are one and the same point. 



ii. If two systems ft, ft' be coresidual, any system a' which 

 is a residual of one will be a residual of the other. 



Suppose that through any system a, two curves U p , U q are 

 described meeting the quartic again in systems ft, ft', then by 

 definition ft, ft' are coresidual systems ; then if through ft' a curve 



* Some interesting results connected with bicircular quartics, obtained by 

 developing this theorem, were given in a paper communicated to the London 

 Mathematical Society in May, 1890. 



