Prof. Sylvester's Astronomical Prolusions. 61 



the vanishing of the coefficients of the terms that ought to rise to 

 the fourth degree in the variables. Calling, then, this norm N, 

 we see that the quadratic equation N=0 is satisfied alike by the 

 equations p + p'=2a and p — p' = 2a, the difference being that 

 one of these will belong to the apex of an actual, and the other 

 to that of an ideal triangle, according to the sign of e— 1. 



It may not be quite out of place here to show how immediately 

 the knowledge of the existence of a third focus to the Cartesian 

 ovals, that remarkable discovery of our illustrious Royal Society 

 Laureate of the year, flows from a similar process to the one above. 

 For taking the norm of the expression 



a Va? + y 2 + VbV + b y + 2bcx + c 2 + d*, 

 the equation to any such curve becomes 



-2d 2 ((& 2 + « 2 )(# 2 + y 2 ) + 2fo# + c 2 ) 

 +<* 4 =0, 



i. e. (6 2 -« 2 ) 2 (^ 2 +y 2 ) 2 +4^(6 2 -« 2 )^ 2 + 2/ 2 ) 



+ (2(c 2 -rf 2 )& 2 -2(c 2 + rf> 2 )(# 2 +y 2 ) 



+ 4& 2 A 2 + 4(c 2 -d 2 )fo# + (c 2 -d 2 ) 2 =0; 



in which equation a 2 , Z> 2 , c 2 , d 2 may obviously be replaced by a 2 , 

 /S 2 , 7 2 , 8 2 , provided 



yS 2 ~« 2 =i 9 ~« 2 , 



£7 —bc 3 



a 2 ^ = <AZ 2 , 



f-&=f—d*. 



Writing for « 2 , Z> 2 , c 2 , d 2 , &c. a^ £ lt c v d v and squaring the 

 second equation, we obtain a symmetrical system of equations, 

 viz. 



for determining a p y@„ y^ S 1# Throwing out the solution a x = a l9 

 ft } =b v y x =c ly B ] =d v only one other solution will be found to 

 exist, which, restoring the original variables, becomes 



- c i-df' P ~ c *-d* ' 



//2_,,2 



ry2 = Z L A2 £2 



6 2 



with the condition that fiy — bc. 



