QUADRATIC VECTORS. 447 



which factors Hke (203) if the vector coefficients are not in the same 

 plane, that is unless SjSs/Sitt = 0. If so, we must either have tt par- 

 allel to /3i, or may take it parallel to /Ss, which is as yet any vector in 

 the tangent plane. If tt is parallel to /Ss we may subtract the term 

 p(aar2 + gxz) from (186). The resulting quadratic vector may be 

 written 



X2Xz\{gi — a)^z — g^o] — xsXig^i + XiX2{cfi3 — a/Si}, (240) 



where c is a constant which cannot vanish. This is in the form (201) 

 and can be factored, since neither c nor g are zero for this case. If tt is 

 parallel to j8i, the tangent plane to (3) ceases to be uniquely deter- 

 mined, contrary to hypothesis. 



If there exists no axis without the tangent plane', u being still zero, 

 we must have S^iwfx = 0; for if not we should have three axis, (not 

 necessarily distinct), aside from /3i. These cannot be distinct and 

 in the tangent plane, nor, as was shown, can they be coincident in that 

 plane. Hence S/Sitt/x = 0. But a quadratic vector of constant plane, 

 having a zero in that plane, and only one other axis in that plane, must 

 have that plane tangent to the cones (3) at the zero.^^ Hence tt lies 

 in the plane X2 = 0, and may be taken to be parallel to 03- Since fi 

 is in the same plane we may take /x = 6i/3i + 63/33. The vector (185) 

 now takes the form 



^3(curiX2 + gxi" + gix^xs) + Wi + bs^3)x2\ (241) 



where a is a non-vanishing constant. 



To see whether a term pSdp can be added to this vector so that it 

 can be factored into V(})f)dp, the most elegant method would be to 

 consider, after Hamilton, the pure and the rotational parts of strains 

 defined by </> and 6. As I have not in the present paper introduced 

 these ideas, I shall employ the more cumbrous method of undeter- 

 mined coefficients; and shall thereby avoid a digression upon simul- 

 taneous forms of (j) and d. Since the vectors |Si, 182, and ^3 are diplanar, 

 a general form of and 6 with undetermined constants p and q may 

 be taken as 



<t>p = Xi(jyV^2^3^qV^3^,+ rV^r^,)+Xo(jj'V^o^3-\-qJ%^i+r'V^^2) 



+.r3(i/' r/32/33+9" F/33^i+ r" F/3,/32), 

 dp = Xi(j>,V^2^3-\-qiV02^i-\-riV^i^2)+X2{p'iV^203-\-q\l%^i 



-}-r\V0^2)+X3(^'\V^2l33^q"iV^3^i-\-r'\V^i^2). 



21 Proved in Art. 35. 



