A Treatise on the Theory of Screws

310 THE THEORY OK SCREWS. [292, its correspondent in the other system will be A/Aj Gtj j ... Ctfi. Similarly, the correspondent to Ä, ßn will have for its co-ordinates phlßii ••• P^nßnt and the correspondent to 7i. ••• 7» will have for its co-ordinates AVi, vhnyn, where X, /z, v, are the constants requisite to make the co-ordinates fulfil the fundamental conditions as to dimensions. We thus compute = p {Pih^ß^ + ... + PnhnUnßn) ; and similarly for the other terms. Whence, by substitution, we find the following equation identically satisfied:— It may be noted that, in a three-system, two liomographies are chiastic when, in the plane representation by points, the double points of the two systems form a triangle which is self-conjugate with respect to the pitch conic. 293. Origin of the formulae of § 281*. Let a be a screw about which a free rigid body is made to twist in consequence of an impulsive wrench administered on some other screw y. Except in the case where a and y are reciprocal, it will always be possible (in many different ways) to design and place a rigid body so that two arbitrarily chosen screws a and y will possess the required relation. Let now ß and £ be two other screws (not reciprocal) : we may consider the question as to whether a rigid body can be designed and placed so that a shall be the instantaneous screw corresponding to t] as an impulsive screw, while ß bears the same relation to It is easy to see that it will not generally be possible for a, ß, y, g to stand in the required relations. For, taking a and ß as given, there are five Proceedings of the Cambridge Phil. Soc. Vol. ix. Part iii. p. 193.