364 THE THEORY OF SCREWS. [334- 



A be a screw in U the impulsive cylindroid, then B will be its instantaneous 

 screw in V the instantaneous cylindroid. 



It is however to be now shown that there are two screws H 1 a,nd H 2 on U, 

 and their correspondents K 1 and K 3 on V, which possess the remarkable 

 characteristic that whether V be the impulsive cylindroid and U the instan 

 taneous cylindroid or vice versa, in either case H l and K l are a pair of corre 

 spondents, and so are H 2 and K z . 



Let B 1} B 2 , B 3 , &c. be the screws on U corresponding severally to the 

 screws A l} A. 2 , A 3 , &c., on V when V is the impulsive cylindroid and U the 

 instantaneous cylindroid. 



Let G l} C 2 , C 3 , &c., be the screws on U corresponding severally to the 

 screws A l} A 2 , A 3 , &c., on V when U is now the impulsive cylindroid, and V 

 the instantaneous cylindroid. 



The systems A lt A 3 , A,, &c., and B,, B.,, B 3 , &c., are homographic. 

 The systems C lt a , C s , &c., and A lt A s , A 3 , &c., are homographic. 



Hence also, 



The systems B lt B. 2 , B 3 , &c., and C lt C*, C 3 , &c. are homographic. 



Let 7/j, HZ be the two double screws on U belonging to this last homo- 

 graphy, then their correspondents K l} K 2 on V will be the same whether U 

 be the impulsive cylindroid and V the instantaneous cylindroid or vice versa. 



There can be no other pairs of screws on the two cylindroids possessing 

 the same property. 



335. A Property of Co-reciprocals. 



Let a, /3, 7 be any three co-reciprocal screws. If 77, ff, are the three 

 screws on which impulsive wrenches would cause a free rigid body to twist 

 about a, /3, 7 respectively, then 



cos (af ) cos (17) cos ( 7 ) + cos (a) cos () cos (77;) = 0. 

 We have from 281 the general formula 



cos fy + cos 



cos a?; cos 



but as a and /3 are reciprocal each side of this equation must be zero. We 

 thus have 



Pa 



cos , = - -- cos 



^ -/ 



cos (017) cos 



