225J FREEDOM OF THE FOURTH ORDER. 235 



degree seems not to be justified. It is however shown in 202 that this 

 hyperboloid is really not more than a part of the locus. There are also four 

 imaginary planes which with the hyperboloid complete the locus, and the 

 combination thus rises to the sixth degree. 



225. The Quadratic Systems of Higher Orders. 



If we had taken n = 3, then of course the quadratic three-system would 

 mean the collection of screws whose four co-ordinates satisfied an equation 

 which in form resembles that of a quadric surface in quadriplanar co 

 ordinates. A definite number of screws belonging to the quadratic three- 

 system can in general be drawn through every point in space. 



We shall first prove that the number of those screws is six. Let l} ..., 8 

 be the co-ordinates of any screw referred to a canonical co-reciprocal system. 

 Then if x , y , z be a point on 0, we have ( 43) 



(0 5 + 0.) y - (0 3 + 4 ) / = a (6, - 0,) - p e (e l + 0,), 



(0, + 0,) z 1 - (0 3 + (i ) x = & (0 3 - 4 ) - Pe (03 + 0.1 



(0 3 + 04) x - (0, + 0,) y = c(0 6 - 6 ) - p e (0 5 + 6 ). 



If we express that belongs to the enclosing four-system we shall have two 

 linear equations to be also satisfied by the co-ordinates of 0. These equations 

 may be written without loss of generality in the form 



2 = ; 4 = 0. 



We have finally the equation U g = characteristic of the quadratic three- 

 system. From these equations the co-ordinates are to be eliminated. But 

 the eliminant of k equations in (k l) independent variables is a homo 

 geneous function of the coefficients of each equation whose order is, in 

 general, equal to the product of the degrees of all the remaining equations*. 

 In the present case, the coefficient of each of the first three equations must 

 be of the second degree in the eliminant and hence, the resulting equation 

 for p e is of the sixth degree, so that we have the following theorem. 



Of the screws which belong to a quadratic three-system, six can be drawn 

 through any point. 



As the enclosing system in this case is of the fourth order, the screws of 

 the enclosing system drawn through any point must lie on a cone of the 

 second degree ( 218). Hence it follows that the six screws just referred to 

 must all lie on the surface of a cone of the second degree. 



We may verify the theorem just proved by the consideration that if the 

 function U 6 could be decomposed into two linear factors, each of those factors 



* Salmon, Modern Higher Algebra, p. 76, 4th Edition (1885). 



