90 Mr Lachlan, On some theorems [May 26, 



then the circle which touches the quartic at Qj and passes through 

 Q will cut the curve again in P the coresidual of five consecutive 

 points at P. Hence we have the theorem that if the bitangent 

 circles at P touch the bicircular quartic again at the points 

 Qj, Q 2 , Q 3 , Q 4 and the osculating circle at P cut the curve again 

 in the point Q, then the four circles which can be drawn passing 

 through Q and touching the quartic at Q v Q 2 , Q 3 , Q 4 cut the 

 quartic again in the same point P. 



Again if we draw a bicircular quartic passing through the five 

 points Pj, P 2 , P 3 , P 4 , P s it must cut the given quartic in three 

 points p % , p 2 , p 3 , the circle through which must pass through P 

 the coresidual of the five given points ; hence if we wish to draw 

 a bicircular quartic passing through five given points on a given 

 bicircular quartic and osculating the latter elsewhere, we have 

 merely to draw a circle osculating the given curve and passing 

 through P the coresidual of the five given points ; but nine such 

 circles can be drawn ; hence nine systems of bicircular quartics 

 can be drawn passing through five given points on a given 

 bicircular quartic, which have three-point contact with it elsewhere. 



9. To find the residual point of a system of seven given points. 

 Let the points be P v P 2 ,...P 7 ; through any three of them such 



as P i; P 2 , P 3 draw a circle cutting the quartic in Q 1} through P 4 , P 5 

 draw a circle cutting the quartic in Q 2 , Q 3 , and through P 6 , P 7 

 a circle cutting the quartic in Q 4 , Q 5 . And then we may find P 

 the coresidual of the system Q v Q 2 , Q 3 , Q 4 , Q 5 as in § 8, P will be 

 the residual of the given system of seven points. 



Or we might replace any six of the points by their coresidual 

 points; thus let the circle P l P 2 P 3 cut the quartic in Q 1} and the 

 circle P i P s P a cut the quartic in Q 2 ; and then let any circle be 

 drawn through Q t , Q 2 to cut the quartic in P^,P 2 ; which two 

 points constitute a coresidual system of the system P t , P 2 ...P 6 . 

 Then if the circle P/, P 2 ', P 7 cut the quartic in R, R will be the 

 residual of the given system. 



By this method we are enabled to find the eighth point in 

 which any bicircular quartic which passes through the seven given 

 points cuts the given quartic, for every bicircular quartic through 

 the seven given points must pass through P. This method 

 assumes that one quartic through the seven points is given ; and 

 thus the problem is not the same as finding the eighth point when 

 only seven points are given. 



10. To find the residual point of seven consecutive points. 

 Let the points P V ..P 7 in §9 coincide with the point P, let the 



circle of curvature at P meet the curve in Q, also let Q 1} Q 2 , Q 3 , Q 4 

 be the points of contact of the bitangent circles at Q; then two 

 consecutive points at Q l may be considered as a coresidual system 



