Contacts of Lines and Surfaces of the Second Order. 121 



tain of them have remained until this hour unknown and un- 

 suspected. 



The method which we shall pursue is an exhaustive one, and 

 will conduct us by a natural order to a systematic arrangement 

 of all the different modes and gradations of such contacts. 



In a paper in this Magazine for November 1850, I explained 

 the decline of minor determinants, and stated a law, called the 

 homaloidal law, concerning them. 



If U and V be characteristics of the two loci whose contacts 

 are to be considered, U + X,V will be the function, the properties 

 of whose complete determinant, and of the minor systems of 

 detenninants belonging to it, will serve to specify the nature of 

 the contact. 



It will be remembered, that, whatever be the number of vari- 

 able letters in any quadratic function U, three of its first minor 

 determinants being zero, make all the first minors zero ; six of 

 its second minors being zero, make all the second minors zero ; 

 and so on for the third, fourth, &c. minor systems according to 

 the progression of the triangular numbers. 



It is well known that whatever linear transformations be ap- 

 plied to a quadi'atic function W, the complete determinant thereof 

 vfiW remain unaltered, except by a multiplier depending upon 

 the coefficients introduced into the equations of transformation ; 

 consequently the roots of X, in the equation obtained by making 

 the determinant of U + A,V zero remain unaffected by such trans- 

 formation ; and any relation or relations of equality among the 

 roots of the equation | | (W + XV)=0 is an immutable property 

 of the system U, V, which is unaffected by linear transformations. 

 Another and more general kind of immutable property (com- 

 prehending the above as a particular case), to which I shall have 

 occasion to refer, is the following. 



Suppose all the minors of any order of W + XV have a factor 

 X + e in common ; this factor Avill continue common to the same 

 system of minors when U and V are simultaneously transformed. 

 This is a very important proposition, and easily demonstrated ; 

 for if X + e be a common factor to all the rth minors of U + XV, 

 (U — eV) will have all its rth minors zero, and therefore, as ex- 

 plained by me in the paper above referred to, U— eVwill be 

 degraded r orders below U or V. This is clearly a property in- 

 dependent of minor transformation, consequently X+e will re- 

 main a factor of the transformed rth minors. 



In like manner it is demon.strablc that any number of distinct 

 factors X -f c, X + e, . . . . common to the rth minors of one form 

 of U + XV, will remain common factors of any other linearly 

 derived form of the same. It is consequently necessary that 

 each rth minor of one form of any quadratic function W shall 



