414 



HITCHCOCK. 



limitation is that neither determinant (123) nor (156) shall vanish. 

 We can evidently number the four single axes in pairs so that this 

 restriction is not violated, for we may not have four distinct axes in 

 one plane. 



22. If, besides having /3i as a triple axis, Fp has also a double axis, 

 the number of possible ways of writing the vector is very large, 

 whether we apply the method of limits, or introduce the vector Qp 

 as in the previous discussion of double axes. As the chief question 

 considered in the present paper is the existence of the various types 

 such that Fp is not reducible, it will be sufficient to note that we may 

 write, for one triple and one double axis, 



. h(P,,P,Pp) , ^ 7^(614) (PePuPp) , ^ 



Pi ~r^ — T^ T^ X 'T P2 ,,^„ST^ r P6 



{PibPePu) 



(126)Z)« 



A(124) (PuP2,Pp) 



(126) (Pl4P25[P4+cPl7]) 



(143) 



obtained from (136) by putting m^2 + n^b instead of 185 and letting n 

 approach zero. Similarly, for a vector with a triple axis and two 

 double axes, we may take. 



. h{P,,Ps, Pp) A(314) (Pz^PuPp) , . /K124) (P14P25PP) 



'^' (P25P36P14) (123)2)36 (123)Z)25 ' 



(144) 



obtained from (143) by writing m/Ss + 7i^e instead of /Se and letting 

 n approach zero. In keeping with the notation already employed, 

 we take D^p = as the polar plane with respect to the cone (137) of 

 the vector /35, i. e. 



D,p = {P,pPuP^) + c[(315) (12p) + (31p) (125)] (123)^ (417) 



with a similar meaning for Dep. Whence we obtain 1)25 and Dz6 by 

 writing 182 and 183, respectively, for p.. 



From their method of derivation, (143) and (144) are the most 

 general vectors of their types. Moreover, (144) is always possible, 

 because the only restriction is that the determinant (123) shall not 

 vanish. But it was shown in Art. 15 that three multiple axes can- 

 not lie in the same plane, i. e. (123) cannot vanish so long as Fp is 

 irreducible. 



The same cannot be said with regard to the above vector (143) 

 because the determinants (123) and (126) must both be different 



