898 Mr. J. J. Sylvester on a remarkable Discovery in the 

 Hence we havej.^raoi Mtii u^ikv^n-, ui won ikde f 



.Ni=:72i v^^mfy''^ '^.^nr% N.r=72i* 'i^itmaA 



where it will be observed that I is the quadratic invariant of U. 



Making now > - 



^i^iitJ-i/'> 5ini>n 72mltiiV^ >^wl t»:w 'lailJuiiw tjljj«9'i & 



iy^^liiiirtia^i the five following equatil&sV-l*'7't"'i« •..M .4 na 



.':i!oii.;;iii u)V.:iinJ Ij/»f.ij10'-.i|i»\; lo VlOOfii .tB3'i2 



4/ ,: - J i 4 ^i^aibnoqadi 



(«4-»')^4 + «6*3-<'6-«2~fl^»«^^*^8«*»^^ ^f I "pdcT 



so that the problem reduces itself to finding v, which is found 

 from the equation of the fifth degree : — 



'0^ 



'li 



*2^ 



'2f 



'3> 



*3 f 



^4-¥. 



a^-^v 



3 



& 



'3^ 



«4 + 



'5> 





4'"^^ 



■-v^^ 





^4-n %ji^^ %; 'T *f5 -'^<^ 







V, it will be observed, being 72 x the quadratic invariant of 



(p^x + qy) {pcfc + q^y) {p^ + q^) (p^x + ?42/). 



when the function is supposed to be thrown under the form of 



^iPi^-^qiyf+70€{p,x-hq,yfx(pc^+q^)^p^x+q.^)^X{p^x+q^)^ 



It is obvious that in the equation for finding v, all the coefii- 

 cients being functions of the invariable quantities p^, q^y &c., 

 and € must be themselves invariants of the given function ; so 

 that the determinant last given will present under one point of 

 view four out of the six invariants belonging to a function of 

 the eighth degree, and these four will be of the degrees 2, 3, 4, 5 

 respectively*. 



* The reasoning in this paragraph seems of doubtful conclusiveness. It 

 may be accepted, however, as a fact of observation confirmed and gene- 

 ralized by the subsequent theorem, that the coefficients are invariants. 



