Contacts of Lines and Surfaces of the Second Order. 135 



contact, in which there is no contact, except only at two points of 

 the right line fomiing part of the complete intersection. 



I belie\'e that this kind of contact, which forms a natiu-al 

 fanuly w4th two others about to be described, and which will close 

 the list, has never before been imagined, and would at first sight 

 have been rejected as impossible. 



Having now exhausted the cases of the fii'st class, in which 

 the minors have no factor in common, and the two sections of 

 the second class, in which the second minors have no common 

 factor, but the fii'st minors of U + X,V a linear or quadratic func- 

 tion of X in common, I descend to the third class, in which the 

 second minors, which ai*e quadratic functions of X, are supposed 

 to have a common factor. 



This common factor must enter twice into each of the first 

 minors by virtue of the law previously indicated, and cannot enter 

 more than tv\ice, as othei'wise the first minor of U + W could 

 only dificr from one another by a numerical multiplier, which is 

 obviously impossible, except when U + XV is of the form {k + X) U, 

 i. e. when the two surfaces coincide. 



Again, the common factor of the first minor must enter three 

 times into the complete determinant ; but there is no reason why 

 it may not enter four times, and thus two cases arise. In the 

 fii'st, the characteristics take the form 



V=w^ + f+z^-{-t^ 



V = ax" + az^ + az^ + i/^. 

 The second determinant having a factor in common, shows that 

 the intersection U, V is contained in a ])air of coincident planes ; 

 but the complete detei'minant, having two distinct factors, evi- 

 dences that these plane intersections, viewed as indefinitely near 

 but still distinct, lie in the same cone, which will be a cone en- 

 veloping both the surfaces U and V all along their mutual inter- 

 sections. This is also seen easily from the forms of U and V ; for 

 we have V— « U = (6 — «)^^, which proves that the intersection 

 lies in the coincident, or, to speak more strictly, consecutive 

 plane /^ = 0; and at any point a? = ^, ?/ = ?;, 2' = f, the tangent 

 plane to each surface becomes 



fa? -f- 7??/ -1-^ = 0. 

 As there arc six independent, i. e. non-necessarily co-evanes- 

 cent second minors, that the second minor system shall all have 

 a connnon factor, implies the satisfaction of five conditions. This 

 species of contact I call curvilineo-indcfinite ; it is, I believe, the 

 only kind of indefinite contact between two surfaces of the second 

 order hitherto taken acc(junt of. 



There is still, however, a higher species of contact, videlicet, 

 when all the four roots of the complete determinant of U-|-XV 

 are identical with the root common to each of its second minors. 



