382 



HITCHCOCK. 



(123)2 (45Fop) = h (PaP,Pp), 

 (123)2 (56Fop) = /C4 {Pf,P,Pp), 

 (123)2 (64Fop) = h {P^PaPp), 



(24) 



To collect results, note the identity (which we may obtain by writing 

 out determinants), 



Zo(456) = hn{5QFop) + &5i(64Fop) + 66i(45Fop), 



where 641, 651 and bei are the first or cc-components of 184, ^5, jSe- If 

 this identity, with two similar identities "^ for Fo and Zo, be multipled 

 respectively by i, j, and k, and the results added, the vectorial identity 

 is obtained 



(456) Fo(p) = /34(56Fop) + i85(64Fop) + ^6(45Fop), (25) 



Values determined by equations (24), (equivalent to the six equations 

 (13), necessary and sufficient that 0i, 185, and jSe shall be axes), intro- 

 duced in (25), give 



(123)^ (456) Fo{p) = kMP^PePp) + kMP^P^Pp) + kMP^PM, 



(26) 



The form of this result shows that, on the one hand Foifii), Fo(fi2), 

 and ^0(183) vanish, (because P(|8i), P(J32), and PiPs) vanish), while on 

 the other hand we have 



k,{P,P,Pe) 



^'^^'^ - ^'' (123)2 (456) ' 



(27) 



with similar expressions for Fo(/35) and FoilSe). As a step in the demon- 

 stration of theorem I, we note that two vectors, alike in having 

 j8i, 182,. . .|86 for axes, can be thrown into the form (26), and will then 

 differ in the constants ki, k^, k^, but not otherwise. 



It remains to dispose of ki, k^ and A'e so that ^^ shall be an axis. Let 

 )37 be expressed in terms of 184, 185, and ^^ by an identity Hke (25), viz. 



. (456) /37 = ^4(567) + /35(647) + ^6(457), 



(28) 



If ^^ is an axis, FoOS?) = h^i where h is some constant; whence, 

 writing ^^ for p in (26), 



7 The three are equivalent to the well-known vector identity, (/S, X, n, v, 

 being any four vectors), pS\txv = \Sfjivl3 + nSi>\^ + vSXuff. 



