22 8 J FREEDOM OF THE FOURTH ORDER. 243 



of the screws in the third degree. We can express this condition as a 

 determinant by employing a canonical system of co-reciprocals. For if two 

 screws 6 and (f&amp;gt; intersect, then there must be some point x, y, z which shall 

 satisfy the six equations ( 43) : 



( 5 + ) ?/ - ( 3 + 4 ) z = a (! - 03) -p a (! + 2 ), 

 (! + a 2 ) z - ( 5 + 6 ) x = b ( 3 - 4 ) - p a (a s 4- 04), 

 (a s + a 4 ) x - (! + 03) y = c ( 5 - 6 ) -p a (o s + a,,), 

 (0 B + 6 ) y - (0, + 0&amp;lt;) z = a (6, - 2 ) -p e (6, + 6&amp;gt; 2 ), 



From these equations we eliminate the five quantities x, y, z, p e , p a and 

 the required condition that and a shall intersect, is given by the equa 

 tion 



- ( 5 + 6 ), - ( 3 + 04), (! + a a ), , a (! - a 2 ) = 0. 



-(, + ,), , (! + &amp;gt;(), (a 3 + a 4 ), , 6(a 3 -a 4 ) 



(a 3 + 4), -(ati + Os), , (a 5 + a fi ), , c (a 5 - a 8 ) 



, (0 5 + 6 ), -(6 3 +6&amp;gt; 4 ), , (6, + e,}, a(0 1 -0 2 ) 



-(0 5 +0 6 ), , (0 1 + 2 \ , (6&amp;gt; 3 + ^ 4 ), b(0 3 -0 4 ) 



(0 3 +0&amp;lt;\ -& + 0J, 0,0, (^ 5 + ^ 6 ), c(0 5 -0 6 ) 



Four homogeneous equations between the co-ordinates of indicate 

 that the corresponding screw lies on a certain ruled surface. Let us suppose 

 that the degrees of these equations are I, m,n,r respectively, then the degree 

 of the ruled surface must not exceed Slmnr. 



For express the condition that shall also intersect some given screw ct, 

 we then obtain a fifth homogeneous equation containing the co-ordinates of 

 in the third degree. The determination of the ratios of the six co-ordi 

 nates 6-1, ... 6 is thus effected by five equations of the several degrees I, m, 

 n, r, 3. For each ratio we obtain a system of values equal in number to 

 the product of the degrees of the equations, i.e. to Slmnr. This is accord 

 ingly a major limit to the number of points in which in general a pierces 

 the surface, that is to say, it is a major limit to the degree of the surface. 

 Of course we might affirm that it was the degree of the surface save for the 

 possibility that through one or more of the points in Avhich a met the surface 

 more than a single generator might pass. 



As an example, we may take the simple case of the cylindroid, in which 

 I, m, n, r being each unity the locus is of the third degree. The screws of 



162 



