QUADRATIC VECTORS. 423 



irXiX2 + T.T3^ + f.r2X3 + fXXi^, (171) 



where the terms in .Ts-ri have been removed as in the preceding article. 

 By the rule of Art. 18, the condition that j8i shall be a triple axis is 

 S/SiTTT = 0, the vectors ir and r agreeing precisely with those of that 

 article. 



The condition that j8i shall be a quadruple axis will appear as a 

 relation between the vector coefficients in (171). As in Art. 17 we 

 may write 



xe = ge (172) 



If we take the space-derivative of both sides in the direction e and 

 then operate by St, we have, remembering that any derivative of the 

 unit-vector e is at right angles to e, 



dhf dhf 



d 

 where the operation — means the same as — SeV. 



dh^ 



The condition for a quadruple axis is that the right member of (173), 



on writing /3i for p after the differentiation, shall be independent of X. 



To obtain the condition in convenient form we have to expand the 



left side in terms of differential operations performed directly on the 



vector VpFp. We have, since x* = ^i~> 



dhf 



$€ — - = Se • — , where v = Ua = UVS\pFp, as in Art. 17, 



dh^^ dh^ ""« T(T 



_ ^ d { 1 da a dTa ) 



^dh,lYa'dh,~T^'~dK) 



„ ( 1 (fV 2 da dT<x , ^ . ) 



= Se i — • • - — • h terms ma)-, 



I Ta dK T'(T dK dK ) 



by the ordinary rules for differentiation. If we now write da = <f)dp, 



so that —— = 06, and, as in the investigation of triple axes, write a 

 dhf 



