296 Mr Richmond, On the Reduction of the General 



On the reduction of the general ternary quintic to Hilbert's 

 canonical form. By H. W. Richmond, M.A., King's College. 



[Received 12 March 1906.] 



The theorem that a ternary quintic form may generally be 

 uniquely expressed as the sum of the fifth powers of seven linear 

 forms is due to Hilbert*: recently the theorem has been re- 

 discovered in ignorance of Hilbert's result by myself, and again 

 a year later by Palatini f. The possibility of reduction to this 

 form being established, a method of effecting the reduction (de- 

 pending upon the solution of a single irreducible equation of the 

 seventh degree) which I obtained soon after the publication of my 

 former paper is here indicated. 



The ternary quintic form is the equation of a plane curve of 

 the fifth order, G s : if italic letters x, y, ..., with suffixes 0, 1, 2 

 be the coordinates of points, and Greek letters stand for coordi- 

 nates of lines, the equation of C 5 may be written 



a x 5 = (a x + a 1 x 1 + a 2 x 2 f = 0, 



while G n and r„ will denote a curve of order n and of class n 

 respectively. It is clear that 



The T 3 's apolar to Q 5 form generally a linear system of four 

 terms ; in fact the only exceptions to this statement are when G 5 

 possesses an apolar T 2 . Among the r 3 's apolar to 5 are included 

 those which touch the seven lines representing each of the seven 

 linear forms mentioned above. It will appear that it is always 

 possible to find three T/s apolar to C 5 whose equations T = 0, 

 T 1 = 0, T 2 = satisfy 



foTo + fcTi + fcT.sO (1), 



the solution being generally -unique ; if so, the three T/s have 

 seven common tangents which give the seven linear forms 

 sought. 



That the identity (1) should hold it is necessary and suffi- 

 cient that 



I o = «T i ^2 ~~ 52 — i j Tj = £ 2 2, — £ 2/ 2 5 T 2 = £ o ^q ~~ fei ^o j 



* LiouvilWs Journal, Series 4, Vol. iv, p. 256. 



f Quarterly Journal of Mathematics, 1902, p. 337. Atti d. R. Ace. d. Sc. di 

 Torino, 30 Nov. 1902 ; Rendic. d. R. Ace. del Lincei, 17 May, 1903. 



