358 Mr Bromwich, The Classification of Conies and Quadrics. 



The Classification of Conies and Quadrics. By T. J. I'A. 

 Bromwich, M.A., Fellow of St John's College, Cambridge. 



[Received 26 April 1900.] 



This problem is one of the oldest in the theory of quadratic 

 forms and has been handled by many writers. Until recently, 

 however, it has not been treated by the Weierstrassian methods 

 of classification, and this is probably due to the fact that some 

 modifications are necessary which prevent the direct application 

 of the method. 



Hensel has attacked the problem (Crelle, 113 (1894), p. 303) 

 without using metrical methods (i.e. without the absolute) and his 

 results agree with what has been found in Sec. 2 of the following, 

 but of course the reduction is not complete from a metrical 

 point of view. Hensel's process is considerably simplified by this 

 omission. 



Timerding (Crelle, 122 (1900), p. 172) has recently published 

 an investigation on the same lines as Hensel's ; this follows very 

 closely the reduction of a quadric to its centre, given in books on 

 solid geometry. 



Gundelfinger (Vorlesungen aus Anal. Geom. Kegelschnitte, 

 edited by Dingeldey, 1895) has considered the problem in all its 

 generality for conies; and Briickel (Crelle, 119 (1898), p. 210 and 

 p. 313) for quadrics. The method given below differs considerably 

 from theirs (which are essentially the same) in the case of point- 

 coordinates. For tangential equations their method is virtually 

 the same as mine (Sec. 3) ; but I have abbreviated the process 

 by using a method of simplifying Weierstrass's solution of the case 

 when one of the fundamental forms of the family (Schaar) has 

 a zero determinant. (See a note presented to the London Math. 

 Soc, 5th April, 1900.) 



It should be remarked that the method of reduction given 

 in Sec. 1 below ought to be capable of being applied to the case 

 when the comparison quadric is not of the special type used in 

 the remainder of the paper (i.e. is not definite). 



