100 Proceedings of the Roijal Irish AcaJeni)/. 



we have at once, since the ^'s are self-conjugate, S^' = 0, 



Vp = 'S.\_<^niSa„i - UnSi<iy„i], /( = 1, 2, 3 



= - <t>iai - <p2a2 - <t>3a3 + vi/'a^ +' m,"a:, + m^a^ = Xif'i <" X-"^ + X»"=' 



iiin" being the first invariant and ;^„ being second associated function of <^„. 



16. The occurrence of Smia/' suggests another function. For this, i.e. for 



Sm/'oi, n = 1, and .•. for xioi « = 1. '•' " = 1 for 2(^i«i. 



17. An absolute invariant is got by taking the quotient of any two 

 invariants where % is of same power in each. Thus the following are 

 absolute invariants : — 



S^iOi S^iOi 2\ai 



the fii'st being a quaternion invariant, the two others vectors. For 

 Sci\a'ia'z = PSuia^aj. V^y = 2 (m"iai + m"-iai + m'^a^. 



18. On the way to the evaluation of V-i^ {pp) we get 



FVi/. (pp) = zs-p: + zs.y + OTjS, 

 where Wi = 



V{i(t>ii + jfjiij + k<l>ik) SXi + V(i<t>2i + j^^j + k<l>Jc)S/ii + V(i<f)3i +jti>^ 



+ lc4Jc) Svi + V{itj>iiS\i + j<f)tiSXj + k<f)iiSXk) + V {i<f>2iSni + j4>-iiSpj 



+ k<i>iiSfik) + V (itpaiSvi + j<^:dSvj + k(f>3iSvk). 



The first three terms vanish for all \, /j,,v; and if we use the particular values 

 of X, fi, z^, as in § 8, roj = also w-, = ^3 = 0. 

 So finally FV)/- (pp) = 0. 

 . (Aj) This may be proved also as follows 



d . Sp-ip (pp) = '3Sdp-\p (pp), 

 for the special values as in § 8. 



•■• ^Sp^ {pp) =- H (pp) ; 

 • .-. V'Spxl.(pp) = -3V^I.{pp); 

 .: rVxp (pp) = 0. 



19. For the function 2 FoiOs F<^ip^2P, we have ?i = 2, since 



2 Va\a; V^\<i>'.,p = P-% Va^a. V4>^p<l>,p. 

 We can operate on this quaternion with V-, but the examples worked out are 

 sufficient. 



20. Functions where n = 3 are 



Spt^i (pp) and S4)ip(l>-2p4>3p- 



