382 HITCHCOCK. 



(123)2 (45Fop) = ^6 (PaP^Pp), 



(123)2 (56Fop) = h {PtPePp), (24) 



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



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

 out determinants), 



Zo(456) = hn{5QFop) + 65i(64Fop) + hi{^5Fop), 



where 641, &51 and ^ei are the first or x-components of ^t, 185, 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) + /35(64Fop) + ^e{4.5Fop), (25) 



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

 (13), necessary and sufficient that 184, 185, and jSe shall be axes), intro- 

 duced in (25), give 



(123)2 (456) Fo{p) = kMP^P^Pp) + JcMP^PaPp) + k^P^P^Pp), 



(26) 



The form of this result shows that, on the one hand Fo(0i), Foifii), 

 and Foi^s) vanish, (because P(|3i), Pifii), and POSs) vanish), while on 

 the other hand we have 



fc4(P4P5Pe) 



^«^^*^-^^- (123)2 (456)' ^-^^ 



with similar expressions for Po(i35) and Foi^e). As a step in the demon- 

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

 /3i, jSo, • • .^6 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 ke so that jS? shall be an axis. Let 

 /St be expressed in terms of ^i, jSs, and jSe by an identity like (25), viz. 



(456) /3; = /34(567) + i35(647) + ^,{457), (28) 



If ^7 is an axis, PoO?;) = hjSj where h is some constant; whence, 

 writing 187 for p in (26), 



7 The three are equivalent to the well-known vector identity, (/3, X, ju> f, 

 being any four vectors), j3»SXm»' = 'KS/j-v^ + /j.SvX^ + vSXuff. 



