94 Proceedings of the Roj/al Irish Academy. 



(Bj) m' = hi^m\ + hSn\ + kihn', + liJ^iii + hkfii, + Jctkjjn, 

 (C|) m" = /iim"i + kim"i + k^m'^. "Where 



.„ , , 2'S0,X(ip2jU^3V + (paflipiv) 



(■Ci; ^123 = ^T , 



OA/XV 



X, /i, v being any three non-coplanar vectors. 

 $' satisfies the same equation. 



2. If we put 



we have 



(Aj) (^123 = X2301 + X3102 + Xl2^3 = l^'l x'^ + f'^X'^' + ^''^ X "' 



For iS\fxvdi23 = '2,S(j)i\\'aV(iv = 2;S'A(i'ix'23^A"'! 



and A, )u, 1' are arbitrary. Evidently x^s = X32- 



The general function whose components are ^.o, (pth, <p3c, may be treated 

 similarly. 



2a. Since 



/S'Xjuv SX/iv 



S (pi\-)(^ itVfiv = aiiSXfxv - S(piX(piii(piv ; 



and .■., since A, /i, v are arbitrary, 



(Aja) Xia = «i2^'r' - 'l'ih<p' C'^ <p' -.(p' i'], 



and similarly 



(Bja) Xi» = f'sii^V' - Bls^V^^'lV'VS 



with analogous results for 0i and <pi. 



Equating these and reducing, we get the cubic of <p',(p\'K 



(Cia) mj [i^'i^V]' - «ai [^'i^V']' + «12 [«^'i^V'] - r/h = 0, 



and the three invariants of ^/^j'"' are 



m,i , 0,2 . aji 



mi2 = — , m ,2 = — , m ,2 = — , 



m, 7?H m2 



