QUADRATIC VECTORS. 447 



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

 plane, that is unless S^^^it = 0. If so, we must either have t par- 

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

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

 p(ax<y + gxi) from (186). The resulting quadratic vector may be 

 written 



X2X3{(gi - a)/33 - 9^2} - .r^xig^y + XiX2{c03 - 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 jSi, 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^nrfj, = 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 t lies 

 in the plane X2 = 0, and may be taken to be parallel to 0s. Since fx 

 is in the same plane we may take m = ^ii^i + '^3183. The vector (185) 

 now takes the form 



^3(cu:ia-2 + gxr + ^i.r2.r3) + (ii/Si + ^3133)0:2^ (241) 



where a is a non-vanishing constant. 



To see whether a term pS8p can be added to this vector so that it 

 can be factored into V4>pQp, the most elegant method would be to 

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

 defined by 4> 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 and 6. Since the vectors /3i, jSq, and (83 are diplanar, 

 a general form of ^ and 6 with undetermined constants y and q may 

 be taken as 



0P = .Ti(pT',3o^3+?r/33/3i+rr/3ij8o)+.ro(/>'I'/32/33+9iF^3/3i+r'F^i,32) 



+a:3(p"F^2/33+9"F|83,8i+r"r'/3i/32), 

 ep =.Ti(iJiF/32/33+gJ%^i+rJ'/3:i32)+.r2(yiF^2i33+(?'iF/33^i 



+ r'iF^i,32)+.r3(i>"iFM3+9"iF|33/3i+/-"iT'/3i,32). 



21 Proved in Art. 35. 



