128 Mr. J. J. Sylvester's Enumeration of the 



Let us begin with considering the cases of contact for which 

 the first minor (and consequently a fortiori the minors inferior 

 to the fii'st) have no factor in common. 



Here I I (V + XU) is a biquadratic function. 



If X have all its roots unequal, we have U and V as above given. 



If two roots are equal, the characteristics assume the form 



\]=x^ + y^ + s'^ + a;t "^ 

 Y = ax'^ + bs'^ + cz^ + da;f j 



The touching plane is a?=0; the point of contact is x = 0, y=0, 

 z = ; the cm-ve of intersection is one of the fourth degree, with 

 a double point at the point of contact. 



There is but one condition to be satisfied, and the contact may 

 be entitled " simple " and of the fii'st degree. 



Next let X have three equal values, the equations become 



Y =as^ + ys + at'^ + bxy. 



The tangent plane at the point of contact y = 0, and the point 

 itself a?=0, y = 0, /=0. The curve of intersection is a curve of 

 the foiu'th order, with a cusp at the point of contact. The 

 number of affirmative conditions to be satisfied is two ; the con- 

 tact is of the second degree, and may be termed " proximal " or 

 cuspidal. 



Next let I j (U + XV) have two pairs of equal roots, we shall 

 find 



1] = x~ + xy -^ st 



V = ayz + bxy + czt. 



The line x=.Q, z=0 mil be common to both surfaces. The 

 curve of intersection will therefore break up into a right line and 

 a line of the third order. 



The fonner will meet the latter in two points, which will be 

 each of them pomts of contact. The contact is therefore di- 

 ploidal ; but as there is another species of diploidal contact to 

 which we shall presently come, it will be expedient to charac- 

 terize each of them by the nature of the intersections of the two 

 surfaces ; accordingly this may be termed unilinear-intersection 

 contact, or more briefly, unilinear contact. 



The number of affirmative conditions to be satisfied being 2, it 

 may be said to be collectively of the second degi-ee, but (obvi- 

 ously ?) the contact at each of the two points is of the nature of 

 simple contact. 



Lastly, let us suppose that all four roots of U + XV are equal : 

 we shall find, as the most simple expressions of the most general 



