210] PLANE REPRESENTATION OF THE THIRD ORDER. 213 



In a certain sense therefore a must intersect each of these four screws, and 

 accordingly the cubic has to pass through the four points. 



To prove that a is a double point we write for brevity 



p = 0^ + e:- + 3 -, R = p&A + p.&A + p 3 a 3 3 , 



Q = Pidi 1 + pA* + P-*0 /, 8 = ^0,+ a,8, + a 3 3 , 

 L = p&S + p 2 aj + p s a 3 2 , H = a/ 2 + a, 2 + a 3 2 , 



and the equation is 



P(2HR-LS)-8QH=0. 



Differentiating with respect to 1} 0. 2 , 3 respectively and equating the results 

 to zero we have 



= 20j [2RH -LS- aHS] + a, [2aPH -LP- HQ], 

 = 20 8 [2RH -LS- bHS] + a 2 [2bPH -LP- HQ], 

 = 20 3 [2RH -L8- cHS] + a 3 [2cPH-LP - HQ]. 



These are satisfied by 6 l = a l , 2 = 2 , 3 = a s which proves that a is a double 

 point. 



The cubic equation is satisfied by the conditions 



=p ] a l 1 +p 2 a 2 2 +p 3 &amp;lt;x z 3 . 



This might have been expected because these equations mean that o and 

 are both reciprocal and rectangular, in which case they must intersect. Thus 

 we obtain the following result : 



If !, a 2 , a s are the co-ordinates of a screw a in the plane representation, 

 then the co-ordinates of the screw which, together with a, constitute the 

 principal screws of a cylindroid of the system are respectively 



Ps-p* Pi- Pi P*-PI 



1 2 3 



The following theorem may also be noted. Among the screws of a three- 

 system which intersect one screw of that system there will generally be two 

 screws of any given pitch. 



For the cubic which indicates by its points the screws that intersect a 

 will cut any pitch conic in general in six points. Four of these are of course 

 the four imaginary points referred to already. The two remaining inter 

 sections indicate the two screws of the pitch appropriate to the conic which 

 intersect a. 



The cubic 



P(2HR-LS)-SQH = Q, 



