440 Proceedings of the Royal Irish Academy. 



Let i^i8, i^28, &c., F;^ be the intensities of seven impulsive wrenches- 

 on the screws 1,2,.. .7, which equilibrate, then we must have 



-^18 _ -^28 _ o^, _ F-i 

 18 28 ' 78 ' 



Similarly, by omitting the first screw, we can have seven impulsive 

 wrenches which equilibrate, where 



hence we have 



Fl2 _ Fi2 _ Fu _ n _ ^ 



12 13 14 'Is 



12 . 38 Fn . F^ 



12 . 28 -^13 • -^2.' 



Let the instantaneous twist velocity corresponding to F^^ be de- 

 noted by Fjg, then, as when seven wrenches equilibrate, the seven 

 corresponding twist velocities must also equilibrate, we must have in 

 the corresponding system, 



12 . 38 _ Fj., Fas 



13.28 ^13 ^28 



But we must have the twist velocity proportional to the impulsive 

 intensity ; hence, from the second pair of screws we have 



F.^: 



r,s: 



■■F,, 



: F,,, 



and from tlie third pair, 









hence we deduce 



F38 : 





■ F^ 



^13 ^28 Fi3 Fos 



and, consequently, the function of the eight impulsive screws, 



T2 . 38 



13 . 28 



must be identical with the same function of the instantaneous screws. 

 It should, however, be remarked, that the impulsive and instan- 

 taneous screws do not exhibit the most general type of two homogra- 

 phic systems. A more special type of homography, and one of very 

 great interest, characterizes the two sets of screws referred to. As 



