A Treatise on the Theory of Screws

254] HOMOGRAPHIC SCREW SYSTEMS. 269 where U is any homogeneous function of the second order in a1,...ae, and where 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 ß be any two screws, and let 3 and </> be their correspondents, then, when a. is reciprocal to </>, ß will be reciprocal to 6. We may, without loss of generality, assume that the screws of reference are co-reciprocal, and in this case the condition that ß and 3 shall be co- reciprocal is Pißi&i + - • • • + Psßß» = 0 ; but by substituting for 0lt... 36, this condition reduces to adU adU . Ä-T- + ... + Ä-T- =0. da da Similarly, the condition that a and </> shall be reciprocal is 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 Ü 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 pa of the screw a is given by the equation (§ 38), p«=pdii+ ■■■ +p6^'. the six screws of reference being co-reciprocals, the function pa must retain the same form after the transformation of the axes. The discriminant of the function U + Xpa 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.