A Treatise on the Theory of Screws

292] DEVELOPMENTS OF THE DYNAMTCAL THEORY. 309 In general, when two pairs of screws are given on two cylindroids, the chiastic homography between the surfaces is determined. If, however, it were possible to determine two chiastic homographies having two pairs in common, then every homography is chiastic, and the cylindroids are normal. Let a, r/ and ß, £ be the two pairs of correspondents, and let y have the correspondents £ and f', then we have ™a{™ßv™yf = ™af™ßi™y>), ™a.('™ßv™yi = ™af™ßi'™y>), whence i.e. the two cylindroids are normal. 292. General Conditions of Chiastic Homography. We shall now discuss the relations of chiastic homography between two systems of screws in the same n-system. The first point to be demonstrated is, that in such a case every pair of the double screws are reciprocal. Take a and ß as two of the double screws, and r] and £ will coincide with them; whence the general condition, ™<4™ßri™yt = ™ai™ßi™yr, becomes ™ai™aß™ßy = ™ aß™ ß{™ay > ™aß (™a(™ßy — ™ay™ß() ~ 0. One or other of these factors must be zero. We have to show that in general it is impossible for ™a(™ßy — ™ay™ßi to vanish. For, take 7 reciprocal to a but not to ß, then wtt7 = 0; but is not zero, and therefore waf would have to be zero; in other words f must be reciprocal to a. But this cannot generally be the case, and hence the other factor must vanish, that is ™*ß = 0. In like manner it can be shown that every pair of the double screws must be reciprocal. Conversely it can be shown that if the double screws of two homo- graphic systems are co-reciprocal, then the homography is chiastic. Let the «-double screws of the two systems be taken as the screws of reference; then if one screw in one system be denoted by the co-ordinates ... an,