Contacts of Lines and Surfaces of the Second Order. 125 



Here only one distinct pair of lines can be drawn to contain 

 the intersections, showing that three out of the fom- points come 

 together. 



This may be termed " Proximal Contact." The number of 

 affirmative conditions to be satisfied is two, and the contact is 

 therefore entitled of the second degree. 



The tangent at the point of contact is y = 0, and the four m- 

 tersections become 



a? = ?/ = 



These may be obtained from the equation V— aU = 0, which 

 gives ?/=0 or z=0; the former implying concurrently with 

 itself a?2 = 0, and the latter yz = 0. 



Thus we obtain three systems, 



and one 



a?=0 z=0, 



corresponding to three consecutive points and the single distinct 



The detei-minant of (U + XV) being only of the third degree 

 in X, we have exhausted the singularities of the system U, V 

 dependent on the form of the complete determinant of U +XV. 



Let now the first minors of U + XV have a factor in common ; 

 this will indicate that U + XV may be made to lose f wo orders 

 by rightly assigning X, in other words, that the intersections of 

 U and V are contained upon a pair of coincident lines. Here it 

 is remarkable that the original forms of U and V reappear, but 

 with a special relation of equality between the coefficients : we 

 shall have, in fact, 



U = a.'2 + 2/2 + ^2 



\=ax^ + ay'^-\-bz^. 

 This gives the law for double, or, as I prefer to call it, diploidal 

 contact*. By vii-tue of the Homaloidal law, we know that if 



* See my remarks on the conditions which express double contact in the 

 Cambridge Journal, Nov. 1850. If n functions, being all zero, be the con- 

 clusion of a fact, but r independent svzygetic equations admit of being 

 formed between these functions, the number of affirmative conditions re- 

 quired is not n, but («— r) ; because the fact may be expressed by^affirmmg 

 (re— r) equations and denying certain others. Thus if P=0, Q=0, R=0, 

 S=0 express a fact, and 



PP'-l-QQ'+RR'+SS'=o 



PP"-l-QQ"+RR"+ss"=o, 



the fact is expressible by affirming P=0, Q=(), and denying R'S"— R"S'-0, 



