Mr. J. J. Sylvester on a new Class of Theorems. 365 



given in the September Number for the particular case of 

 w = l. 



When 7i = 0, we revert to Mr. Boole's theorem of elimina- 

 tion between U and V already adverted to. The proof, it 

 will be easily recognized, does not require the application of 

 the more general theorem relative to the simultaneous de- 

 pression of orders of two quadratic functions, but only the 

 limited one before given, which supplies the conditions of their 

 simultaneous disparition. 1 now proceed to develope more 

 particularly certain analogies between the theory of the mutual 

 contacts of two conies, and that of the tangencies to the inter- 

 section of two conoids. 



But here again I must anticipate some of the results which 

 will be given in my forthcoming memoir on Determinants and 

 Quadratic Functions, by explaining what is to be understood 

 by minor determinants, and the relation in which they stand 

 to the complete determinant in which they are included. This 

 preliminary explanation, and the statement of the analogies, 

 above alluded to, will constitute my Third and last addition* 



Imagine any determinant set out under the form of a square 

 array of terms. This square may be considered as divisible 

 into lines and columns. Now conceive any one line and any 

 one column to be struck out, we get in this way a square, one 

 term less in breadth and depth than the original square ; and 

 by varying in every possible manner the selection of the line 

 and column excluded, we obtain, supposing the original square 

 to consist of n lines and n columns, n 2 such minor squares, 

 each of which will represent what I term a First Minor Deter- 

 minant relative to the principal or complete determinant. 

 Now suppose two lines and two columns struck out from the 



original square, we shall obtain a system of J - — > 



square?, each two terms lower than the principal square, and 

 representing a determinant of one lower order than those 

 above referred to. These constitute what I term a system of 

 Second Minor Determinants ; and so in general we can form 

 a system of rth minor determinants by the exclusion of/* lines 

 and r columns, and such system in general will contain 



fn.(n—l) (n— r+ 1)1 



L 1.2 r J 



distinct determinants. 



I say Hn general? because if the principal determinant h& 

 totally or partially symmetrical in respect to either or each of 

 its diagonals, the number of distinct determinants appertaining 

 to each system of minors will undergo a material diminution, 

 which is easily calculable. 



