ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SUEEACES. 
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coordinates of P,, we shall have an equation in x,y,.. which will determine a curve 
upon U at every point of which (Pj being taken arbitrarily) it is possible to draw a 
quadric having three-pointed contact. The degree of the curve would appear to be 
3(^—1) 3 w(m+1) 2 =9w(w + 1)(% 2 — 1). 
But regarding the conditions in question as relations between the coefficients of U, 
we have three equations of the degree 4 and one of the degree 1. Hence we may con- 
clude that through any two points in space we may describe 64 surfaces, whose equations 
contain 5 independent constants ( e . g. quartic scrolls having twisted cubics for their 
nodal lines), such that a quadric may be drawn touching them in two points each, and 
having three-pointic contact at one of the two points. 
This theorem admits of obvious generalizations ; but, having reference to the provi- 
sional nature of the numerical results, it seems hardly worth while to make a statement 
of the theorems which will probably require qualification hereafter. 
There is yet another way in which the equations of condition may be regarded. The 
conditions for three-pointic contact involve the coordinates of four points, P, P„ P 2 , P 3 ; 
i. c. twelve disposable quantities. These may be determined so as to satisfy twelve 
equations. Hence it appears that on a given surface we may take four points (4 equa- 
tions), such that a quadric may be drawn touching the surface at the four points 
(2 equations), and having three-pointic contact at two of them (6 equations) ; i. e. 
44-2 + 6 =12 equations in all. 
Again, the conditions for four-pointic contact involve the coordinates of five points, 
P, P 1? . . P 4 ; i. e. fifteen disposable quantities. Hence we may conclude that on a given 
surface we may take five points (5 equations), such that a quadric may be drawn, and 
having three-pointic contact at three of them (l + 3x 3=10 equations) ; i. e. 5 + 10 = 15 
equations in all. 
Or, again, on a given surface we may take five points (5 equations), such that a quadric 
may be drawn touching the surface in the five points (3 equations), and having four- 
pointic contact at one of them (3 + 4 equations); i. e. 5 + 3 + 3 + 4=15 equations 
in all. 
To these theorems others might doubtless be added. 
§ 4. On Points of Four-, Five-, Six-pointic Contact with a Quadric Surface. 
If, setting aside for the moment the question of multiple contact, we fix our attention 
upon a single point P, the formulae established in the preceding section suggest certain 
conditions necessary for the existence on a surface of points of four-, five-, six-branch 
contact with a quadric. These may be described as the conditions for a quartitactic, 
quintactic, sextactic point on a surface. Now, referring to the equations (4), (6), and 
(7) of § 3, and to the process there used for the elimination of the quantities relating to 
the quadric V ( i . e. Q, 12, 13, . .), it appears that, when the contact subsists for a single 
point only, we have available for the purposes of elimination the relations 
0=01 : 0” _ T = 02 : 0 li-1 2= . . , 
