34 Proceedings of the Royal Irish Acadenii). 



but the operand X is any vector whatever, hence 



y - I - TT^ = 0, identically. (35) 



This agrees with (24), and shows « = x" . Finally, form the invariant x" by (81), 

 putting for ^ its value from (27), and we have 



= m {S<pye'-' FA/x + . . .) 



= S^ve\9^ - S<p\0fiB,- - S^uBvdX by (17) ; (36) 



but this makes x" agree precisely with Joly's invariant /'^ in Trans. R. T. A., 

 vol. XXX, part XVIII (March, 1896), p. 713. Inserting the values of x" and 5, 

 (35; gives 



•n-tf = /', - me^-t, (37) 



and by multiplying both sides into fl"' 



JT = ^e-i - //i0 »^r' ; (38) 



whence by taking conjugates 



. „' = /',^-'- „,e'->'r-', (39) 



so that (11) becomes 



r^e^ + r«a0/3 - [/',^-' - m»'-'f ff'-]Fo/3, (40) 



and the problem proposed in Art. 1 is solved. 



It is evident that the right side of this result is, in form, not symmetrical 

 in the two functions <p and 0, while the left is so. Therefore, if wc had 

 interchanged ^ and throughout the investigation, we should have found 



VfaOfi* VBa^ii = [l,<f,'-' - p<p'-<9'<i>'-']Fa0, (41) 



where /, is Joly's invariant defined by 



/.&A^r = :£,S<t>X<)>^0y. (42) 



and p stands for the third invariant of ^. Since the two quantities in brackets 

 in (40) and (41) must W equal, we have the interesting identity connecting 

 li and /'i 



, = r,e-' - }<>e-^4,e-' = /.«-' - p<t>'e<t>-'. (43) 



As a check on thi^ work, we may note that (40) and (41) are generalisa- 

 tions of Hamilton's relation already mentioned in Art 1. Therefore if, as a 

 special case, 9p - p, identically, i.e.. tf = 1, Iwth (40) and (41) must reduce to 



V<t>aii - Va<t>ti = [p" - «-] Vali. (44) 



In the case of (40) this is all but evident; in (41) the reduction follows by 

 the use of the cubic in <f,. The proof is left to the reader. 



