36 Proceedings of the Royal Irish Academy. 



so that (11) now becomes 



V,padfi + VOa^fi = \_U - w'>' - i^'B' + &<{>' + <t>'9'] Voft. (-48) 



The function in brackets must, of course, be equal to the coiresponding 

 expressions in (40; and in (41), and must reduce to p" - <(,' when = 1, as is 

 easily seen 



7. Proof by use of V. 



In what precedes our work has consisted essentially in the solution of the 

 equation (14) by two different methods, first by identities of the form (21) 

 second by using Hamilton's relation for distributing <t, and d. A third 

 method, oflering certain advantages, is afforded by Hamilton's operator V. It 

 is known that if ;^ be any linear vector function, we have 



{xf,- ■^')^ = - VftV\'iP>T, (49) 



where V acts on a, but not on the constituents of \p. Now by comparing 

 (14) with f9) we have 



li - r)^ - ./F,.fl/3 4 9'Vp<pi3. (50) 



whence by (49) 



0' V/.eii + d' Vpf^i = - Vji rv[^' FpOa . 0" Fp^a], (51) 



where V acts on a but not on p. Thus at onct- 



TTp = - rV[^' Vpea + 0' Fjo^-t], (52) 



which givfs TT more directly than the former methods, but leaves the operation 

 V t<:i l>e perfonne«l. It is not difficult to obtain (45) by the application of tlie 

 properties of V. 



8. Comparison iriih Cartesian methods. 



It is highly instructive tti compare identities obtained by the compact 

 and elegant methods of Hamilton with their equivalent in the langui^e 

 of ordinary scalar al;;ebra. Space forbids doing this in general, but as a 

 single illustration let us see what (48) becomes when thus translated. 

 Let (^ and fl be defined by the respective matrices 



P.u P... P» Qu. Qu. Qn, 



^ = P», P». /'» ^ = Qr.. <?«- e«. 



P,>. /'„, P. Q», Q», Q«, 



and let the components of a and /3 be a,, a,, a^ and b,, 6,, &> 

 The vector ^a will then have the components 



Piifli + Piz"7 + ^i/'j ; ^»«i + ■Pjj^i + •Pi/'i; -Pii«i + Path + Paffj; 



and 0^ will have the components 



