165 



II. "A Fourth Memoir upon Quantics." By ARTHUR CA.YLEY, 

 Esq., F.R.S. Received February 11, 1858. 



(Abstract.) 



The object of the present memoir is the farther development of the 

 theory of binary quantics ; it should therefore have preceded so much 

 of my third memoir, vol. cxlvii. (1857), p. 627, as relates to ter- 

 nary quadrics and cubics. The paragraphs are numbered continu- 

 ously with those of the former memoirs. The first three paragraphs, 

 Nos. 62 to 64, relate to quantics of the general form ( * jf^>y, )** 

 and they are intended to complete the series of definitions and expla- 

 nations given in Nos. 54 to 61 of my third memoir ; Nos. 68 to 71, 

 although introduced in reference to binary quantics, relate or may be 

 considered as relating to quantics of the like general form. But 

 with these exceptions the memoir relates to binary quantics of any 

 order whatever : viz. Nos. 65 to 80 relate to the covariants and inva- 

 riants of the degrees 2, 3, and 4 ; Nos. 81 and 82 (which are intro- 

 duced somewhat parenthetically) contain the explanation of a pro- 

 cess for the calculation of the invariant called the discriminant ; Nos. 

 83 to 85 contain the definitions of the catalecticant, the lambdaic and 

 the canonisant, which are functions occurring in Prof. Sylvester's 

 theory of the reduction of a binary quantic to its canonical form ; and 

 Nos. 86 to 91 contain the definitions of certain covariants or other 

 derivatives connected with Bezout's abbreviated method of elimina- 

 tion, due for the most part to Professor Sylvester, and which are 

 called Bezoutiants, Cobezoutiants, &c. I have not in the present 

 memoir in any wise considered the theories to which the catalecticant 

 &c. and the other covariants and derivatives just referred to relate ; 

 the design is to point out and precisely define the different covariants 

 or other derivatives which have hitherto presented themselves in 

 theories relating to binary quantics, and so to complete, as far as may 

 be, the explanation of the terminology of this part of the subject. 



