114 



Proceedings of the Royal Irish Academy. 



In the case of Art. 51, tlie inTariants of i^and of /are connected 

 by the relation 



%F(^i,) . i, = %f{iC) i, + % r,i,, . i, = 2/(h) h + {rn. - 1) e, 

 since V^ VJie. . «\ = 0, and ^ Foj^e . i^ = - eS^i' + ^iiSeii 



It thus appears, by taking 



m -\ 



that Fi2i^(/i) ^"i may be reduced to zero ; and also that F32/(/i) h is 

 independent of the position of the base-j)oint. 



53. The process sketched in the last article is extremely fertile in 

 the formation of invariants, and in the discussion of the properties of 

 linear functions. 



It may be stated generally that, if /i/o . . . /,„ are any linear and 

 distributive functions of quantities qi^z . . . qmi ^^i-Q quotient 



fi^iJi^z-'-fiSm qi,q2"'q„ 



Q = /2^i,/2?2 . • ./22'm ^ ?i) ^2 . . ■ q,, 



is an invariant in so far that the quantities q may be replaced by any 

 linear functions of them Tvith scalar coefficients. 



With particular reference to the Theory of Screws, we may select 

 any number of screws 



(o-iOi), (o-oOo) . . . (o-.v^v), 

 and we may derive the set of invariants of the type 



CTi, 0-2 . 



Oi, O2 



^1,02 



0-iV 





0-3- 





Qn 





^N 





iil, 1*2 



fiiV 



Ol, fi, • • • ^N 



in the dividend, J/ rows of o- being followed by iV- Jfrows of S2, 

 and the divisor being formed by -ZV rows of Q.} 



1 In particular, for a pair of scre'^s on a cylindroid in tKree dimensions, the 

 ratios 



(Tiffo — Caff] : ff\tii'> — crjcoi : wia)2 — wioii 



are independent of the particular pair of screws chosen. 



