QUADRATIC VECTORS. ~ 451 



for a (|iiadruple axis and /So for a triple axis. If we add tlie vector 

 term 



— p{axi + gixs) 



the resulting quadratic A^ector may be written 



i8i(M.r2.r3 - axi") + (g^, - firi^s).^,^ + (- ffi^i - afiz)xzXu (246) 



and if we then take 



<f>P^ - axi{gil%^i + gVl^il^^) - ax2uV^3l3i 



+ xsiga l'%0, - gi-'l'Mi - ggxV^x^^), 

 dp = axj%0, + X3(gJ%0, + r/I"^,^.,), (247) 



we find (24(5) identically ccpial to \'(f)p9p, aside from a scalar factor 

 gaS^i^o^s. 



If no axis exists except in the plane Xo = 0, (having now by hypo- 

 thesis Sirid^i = and u not zero), we must, as before, have x an axis 

 which we may take as ^3. The quadratic vector may then be written 



^3{axiX2 + f/.r3- -r- fifi-fa-fs) + 11^1X2X3 + (bi8i + ^s/Ss).^^'^ (248) 



where the constants a, h\, and 63, have the same meaning as in (241). 



To see whether a term pSbp can be found which shall render this 

 vector factorizable, we may set up equations as for (241), with the 

 difference that the right member of 10 will now be u instead of zero. 

 In the equations under (241), for Cases 1 and 3, equation 10 was not 

 used. Hence the reasoning still holds. For case 2, the reasoning is 

 as before, up to the substitution in equation 10. The result gives 



y = — ^ as a imique solution. The values of 4>p and of dp which follow 



u 



are, on letting p' , (which, from the equations, is arbitrary), have the 

 value —1, and clearing of fractions, 



(/)p = —gX2V^2^3^{gXi — llX2)Vfi3^l, 



Op = (igux2V82&z-\- \{(iu--\rh3gii)x2-^ {gg^u — hig'^)x3] VBz^i 



-{-{b,g'X2-\-ghiX3]l%fi2; 



while (248), after subtracting the term p(yx2-\-gx3), becomes 



^i{ux2X3-\-biX2''—yxiX2—gxiX2)-{-^2(.—yx2^—gxiX3)-\-^3{axiX2-\-giX2X3 



-^h3X2^-yx2X3). (250) 



