310 THE THEORY OF SCREWS. [292, 



its correspondent in the other system will be 



XAj !, ... \h n ct n . 

 Similarly, the correspondent to 



&, ... /3 n 

 will have for its co-ordinates 



Mi/Si, Mn/3n, 



and the correspondent to 



7i&amp;gt; 7n 

 will have for its co-ordinates 



where A,, /n, v, are the constants requisite to make the co-ordinates fulfil the 

 fundamental conditions as to dimensions. 



We thus compute 



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 homographies 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 77. 

 Except in the case where a and 77 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 77 will possess the required relation. 



Let now /3 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 77 as an impulsive screw, 

 while ft bears the same relation to . 



It is easy to see that it will not generally be possible for a, /3, 77, to 

 stand in the required relations. For, taking a and /3 as given, there are five 



* Proceedings of the Cambridge Phil. Soc. Vol. ix. Part iii. p. 193. 



