10 Proceedings of the Royal Irish Academy. 



Finally, from (14), forming the product V(ppOp we have 



V</, P 9 P = SHpVVatFpX + Si P Sjp[VVaeVfo + VV an vpX] + &]pVVm,Vfa. 



(18) 

 Since i and j are at right angles, it is only needful to show that coplanarity 

 of the vector coefficients entails the reducibility of the expression ; in fact, to 

 show that whenever we have 



8 . VV.n F/3A[ Wat Vfifl+VJ \,„ I r JA] VVm, V(Sp = 0, (19) 



we also have the right member of (18) divisible by a scalar factor. The 

 scalar product may be expanded as 



- 8 . Vat V(3\ VfipS . Vat Va„ Vfip + 8 . Vat I )-}\ J',,„S . P/3A J'„ H Y$p, (20) 



by applying the identity 



8 . I i>)p. ' /<7<. Vpsp ■..,■- p. - Sp^ptSpsptpt 



in which we write p, = Vat. p- = F/-JA. &e. Expanding again in a similar 

 manner (20) may be written 



- SatfiSXfiu . (- Sati/Sa/3/i) Sai»jSa/3A . S/3A/U Sai|0. (21) 



Now if *SA/3/« = 0, the direction of 0p in (14) - ■ onstant, hence 0/u is divisible 

 by a linear scalar and 1 18) is reducible. Similarly, if Satij 0, «,, is reducible. 

 Rejecting these factors from (21) we have 



?a ,->...-;„ - SafiXSatifi ' 0, (22) 



as the only re m ain in g possibility. But this is the condition that $//3 shall 

 be parallel to Q' a : for, remembering that 13) is the conjugate of the present. 

 case, we have 



-- iSae/3 Sanfc » ■■ S/3An iSBfia ; (23) 



the condition that the coefficient* oi i and , shall be in proportion is 

 equivalent to (22). W; g „ = .^'^, and take 7 any 



vector not coplanai with a and J. Identically 



whence, operating with 0' and 0', and remembering <p'u = 0^3 = 0, 



(p'pSafiy - f'PSyap - +'yS a ,ip, WpSafa - s^'fiSflyp + U'y8ufip. 



By taking coi g if both si 



fpSa/iy = F- „>.,,,-,„ ra/35*'7/». 'V' s '»/37 = *F]3yfy'/3p+ ValiSO'yp. 

 Forming the vector product we I 



F^pO.S^-y 



= 8<p't3p [x V Vy„ Vp- v\ v„ VafiSffyp + W'I r o/3 VpyS+'yp] 



which contains the factor Sf'fip, hence is reducible. The theorem is 

 therefore proved. 



