42 Important New Theorem. 



quadric (2) measured on any two parallel lines drawn through in 

 and o to pierce the quadric (2). And since II + k V = O, we 



m m 



may put this under the form 



(k'— k) V 



U + k' V 



(4) 



Similarly, the ratio of the powers of m and o' in respect to the 

 quadric (1), measured on any two parallel lines through m and 

 o' to pierce the quadric (1) may be expressed in the form 



(k" — k) V 



IT + k" Y (5) 



o' o' 



Now we easily obtain the ratio of 4 and 5 in the form 



/v i r u + k " v ) 



(raW-fn^} (6) 



^» o J 



which is independent of the position of the point m in the quad- 

 ric (3), and constant for all points in the same. 



Hence the following 



Theorem. "When three quadrics C, C, ^ , have common in- 

 tersection, if from a point m in one of them, we draw any two 

 lines to piex'ce the others C, C in pairs of points a, a' and b, b', 

 and that from any two fixed points o, o' in space we draw paral- 

 lels to these lines to pierce the same quadrics in pairs of points 

 m a . m a' m b . m b' 



a, a and /3, , then will I = constant. 



o a . o a 6 /3 . 6 



And reciprocally. Given two quadrics C, C , and two fixed 

 points o, o' ; if from a point m any two straight lines be drawn to 

 pierce the quadrics C and C' in pairs of points a, a' and b, b', and 

 that parallels to these lines be drawn through o, and o' to pierce 

 the quadrics in pairs of points a, a and /3, ; then should the 



m a . m a' m b m b' 



point m be such as to fulfil the relation '. = 



o a . o a o'0.60 

 a constant ; its locus is a quadric ^ having common intersection 

 with the two quadrics C and C 



Note. — In a supplementary Paper I will show how the developments of 

 these theorems make known to us a new class of truths relating to quadric 

 surfaces, analagous to those (for plane conies) constituting the xvi chapter 

 of Professor Chasles' " Traite cles Sections Coniques." They will give also 

 the principal theorems concerning the '• umbilical " and " modular foci," 

 which have been discovered by Professors MacCullagh and Townsend. 



