A Treatise on the Theory of Screws

___________ _____ _________________ _______________ 314 ___________ __________________ THE THEORY OF SCREWS. [295, The calculation* presents no difficulty and the result is as follows:— + = cos#cos</> +p« cos (/37?)[OTa, cos (/3£) — cos (/3t;)] + Pß cos (ai?) [®-a, cos (a£) - ■nroS cos (a??)] [cos (af) cos (^) - cos (arj) cos (££)] _ + cos 3 sin </> T+ pa cos (8f) |>a, cos (/?£) - cos (/fy)] ' + Pß cos (a£) [•srai cos (a£) — cos (a,;)] [cos (a£) cos (ßif) — cos (a-??) cos (ßl;)] _ + sin 3 cos </> r+ pß cos (ai?) [13-3, cos (a£) — cos (a?;)] + pa cos (ßq) [isrßl cos (ßg) - nrßt cos (^)J _+Pß^an [cos (a£) cos (/fy) — cos (a??) cos (/3£)]_ + sin 3 sin </> [~+ pß cos (a£) cos (a£) - vrß( cos (<w/)] + pa COS (ߣ) [uj-ß, COS (ßl;) — tSßt cos (ßrj)\ ,+ Pß^ai [cos (a£) cos (ßv) - cos (av) cos (ߣ)] _ 296. An exceptional Case. A few remarks should be made on the failure of the correspondence when the principal planes of the two cylindroids are at right angles (§ 294). It will be noted that though this equation suffers a slight reduction when the principal planes of the two cylindroids are at right angles yet it does not become ©vanescent or impossible. For any value of 3 defining1 a screw on one cylindroid, the equation provides a value of </> for the correspondent on the other cylindroid. Thus we seem to meet with a contradiction, for while the argument of § 294 shows that in such a case the homography is impossible, yet the homographic equation seemed to show that it was possible and indeed fixed the pairs of correspondents with absolute definiteness. It is certainly true that if two cylindroids A and P admit of the cor- relation of their screws into pairs whereof those on P are impulsive screws and those 011 A are instantaneous screws, the pairs of screws by which the homographic equation is satisfied will stand to each other in the desired relation. If, however, the screws on two cylindroids be correlated into pairs in accordance with the indications of the homographic equation, though it will generally be true that there may be corresponding impulsive screws and instantaneous screws, yet in the case where the principal planes of the cylindroids are at right angles no such inference can be drawn. The case is a somewhat curious one. It will be seen that the calculation * See Trans. Roy. Irish Acad. Vol. xxx. p. 112 (1894). ______ _________________________ _______ _______________