446 Proceedings of the Royal Irish Academy. 



receives an impulsive wrench on the same screw. These screws are the 

 principal screws of inertia. 



"We have already seen the anharmonic equality between four screws 

 on a cylindroid, and the four corresponding screws ; we have also 

 shown a sort of quasi anharmonic equality between any eight screws 

 in space and their correspondents. More generally, any n + 2 screws 

 of an w-system are connected with their n + 2 correspondents, by re- 

 lations which are analogous to anharmonic properties. The inva- 

 riants are not generally so simple as in the 8-screw case, but we may 

 state them, at all events, for the case of » = 3. 



Pive screws belonging to a 3-system, and their five correspondents 

 are so related, that, given nine of them, the tenth is immediately de- 

 termined ; for this two data are required, that being the number re- 

 quired to specify a screw already known to belong to a given 3-system. 



We may, as before, denote by 12 the condition that the screws 

 3, 4, 5 shall be co-cylindroidal. This, indeed, requires no less than 

 four distinct conditions, yet, as pointed out {Screws, p. 44), functions 

 can be found whose evanescence will supply all that is necessary. JS^or 

 need this cause any surprise, when it is remembered that the evanes- 

 cence of the sine of an angle between two lines contains the two 

 conditions necessary that the direction cosines are identical. The 

 function 



12 . 34 



13 . 24 



can then be shown to be an invariant which retains its value unaltered 

 when we pass from one set of five screws in a 3-system to the corre- 

 sponding set in the other system. When two invariants are known, 

 the required screw is determined. 



