254] HOMOGRAPHIC SCEEW SYSTEMS. 269 



where U is any homogeneous function of the second order in a lt ...a 6 , and 

 where p lt ...p 6 are the pitches of the screws of reference, then the two 

 systems are related by the special type of homography to which I have 

 referred. 



The fundamental property of the two special homographic systems is 

 thus stated : 



Let a and /3 be any two screws, and let 6 and &amp;lt;j&amp;gt; be their correspondents, 

 then, when a is reciprocal to &amp;lt;j&amp;gt;, /3 will be reciprocal to 9. 



We may, without loss of generality, assume that the screws of reference 

 are co-reciprocal, and in this case the condition that ft and 6 shall be co- 

 reciprocal is 



= ; 



but by substituting for 1} ... 6 , this condition reduces to 



dU 



Similarly, the condition that a and &amp;lt; shall be reciprocal is 



dU dU 



~ = 



It is obvious that as U is a homogeneous function of the second degree, 

 these two conditions are identical, and the required property has been 

 proved. 



254. Reduction to a Canonical form. 



It is easily shown that by suitable choice of the screws of reference the 

 function U may, in general, be reduced to the sum of six square terms. We 

 now proceed to show that this reduction is generally possible, while still 

 retaining six co-reciprocals for the screws of reference. 



The pitch p a of the screw a is given by the equation ( 38), 



the six screws of reference being co-reciprocals, the function p a must retain 

 the same form after the transformation of the axes. The discriminant of 

 the function 



equated to zero will give six values of X ; these values of X will determine 



the coefficients of U in the required form. I do not, however, enter further 



into the discussion of this question, which belongs to the general theory of 

 linear transformations. 



