QUADRATIC VECTORS. 451 



for u (luadruple axis and ^2 for a triple axis. If we add the vector 

 term 



— p(a.i-i + giXs) 



the resulting quadratic vector may be written 



0i{ux2X3 - a.vi2) + (5r/3, - g.^^Jxi^ + (- g,^, - a^sjxzxr, (246) 



and if we then take 



<^P = - a.ri((7iI'/33/3i + gV^i^o) - aXiuV^sjSi 



+ xziga F^2/33 - gi^l'%0i - ggx\%^2), 

 Op = (u-J%3i + X3(gJ%l3, + (/F/Si/So), (247) 



we find (246) identically equal to V<f)pdp, aside from a scalar factor 



If no axis exists except in the plane x-i — 0, (having now by hypo- 

 thesis Sir/jL^i = and u not zero), we must, as before, have tt an axis 

 which we may take as 183. The quadratic vector may then be written 



fisiaxixo + gx,^ -{- giX.Xs) + 11^1X2X3 + {bi8, + ba^sjxi^, (248) 



where the constants a, hi, 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 

 7 



V = -^ as a unique 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, 



4>P = - gx2 V^2&z-\- (gxi — nxo) V^^^u 



dp = (igux2Vl3203-{-{iau--\-b3gu)xn-\-(ggiU—big^)x3}V8zpi 



-\-{big'x2-\-ghiX3}V^,^o; 



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



8i{ux2X3-{-b ixr — yxiX2 — gxiX2) +/32( — yx2^ — gx2X3) -\-^s iaxiX2-\-g i.r2^-3 



-\-bzX2^-yx2X3). (250) 



