410 HITCHCOCK. 



Fp - p{AioXs + AnX2), (128) 



namely, the vector into which /3i is multiplied in (125), Syp vanishing. 

 By (9), the substitution of p for (127) is equivalent to 



xi =0, X2 = c'2^22 + c'3^32, X3 = — c'iAiz — c'3^33. (129) 



This gives, for Fa, 



Fa = - (c'2^22 + C'3^32) (c'2^23 + C'3^33) (^21/32 + ^3ljS3), 



neglecting the /3i component. Substituting values in (128), and col- 

 lecting coefficients, the condition SjSittt = may be arranged as 



CV {02(^23^^12 - A2ZA13A22 + ^31^22^23) — 03(^22^13 — A22A12A23 '^ 



+^21^22^23)} 



+ C'2C'3 {02(^22^33 + ^23^32) (+ -431 — ^13) + 2A23A12A23 



— C3(-422^33 + ^23^32) (+^21 — ^12) + 2^122^32^13} 

 + CV {02(^33^^12 — ^33^13^32 + ^31^32^33) — 03(^32^13 — ^ 32 vl 12^33 » 



+^21^32^33)} =0. (130) l 



The three expressions in braces are easily seen to be equivalent, when ' 

 none of the four elements of (63) are zero, in virtue of the equations j 



C2 .<422 ^23 (^'^^^ : 



C3 A32 A 



33 



In any case, all three expressions in braces must vanish, since the 

 choice of constants c'2 and c'3 is arbitrary. 



Considering various cases that may arise, the vanishing of any one 

 of the four elements of (63) entails the vanishing of one of the con- 

 stants C2 or C3, provided the vector Fp is not reducible. For example, 

 suppose ^33 = 0. The vanishing of (63) entails A23A32 = 0. If 

 A23 = 0, the vanishing of the third line of (130) gives y4i3 = and Fp 

 is reducible; whence we have ^32 = and so C3 = 0. Similarly we 

 may show that the vanishing of either ^32, -422, or -423, entails the 

 vanishing of one of the constants C2 or C3. Accordingly we have only 

 three possibilities, — 



(a). Neither C2 nor C3 is zero. The three expressions in braces are 

 equivalent. 



i 



