QUADRATIC VECTORS. 385 



changing the two sets of axes numbered 1, 2, 3 and 4, 5, 6, whence 

 (33) becomes 



(237) -04562 (3, p) (317)-C4562(1,P) ,34. 



'"^^ (345) (236) (247) '^^ (145) (631) (347) ^ ' 



and the two vectors (33) and (34) can, by theorem I, differ only in a 

 scalar factor and a term (px + ??/ + rz)p. As an example of the 

 striking relations that hold between the constants p, q, r and the axes, 

 let p, q, and r be determined so that tp added to (33) gives a vector 

 equal or parallel to (34). If we write 



b = ip -\- jq + kr (35) 



the vector 5 thus determined is at right angles to both the axes ^3 and 

 /Se which do not enter into either of the coplanar sets, a consequence of 

 the fact that ^z and /Se are zeros of both vectors (33) and (34). More 

 generally, if Fi{p) = hF^ip) + tp, and if Fi and Fo have a common zero 

 j8, t must vanish if p has the direction /3, i. e. 5 is at right angles to /3.^ 



8 Darboiix has pointed out, (loc. cit.) the importance of Hnear relations of 

 the type (127) = in the solution of differential equations. For example, 

 that the solution of the equation 



{ijZ - zY)dx + {zX - xZ)dij + {xY - yX)dz = 

 may be made to depend on that of a Riccati equation, it is necessary and suffi- 

 cient that we have, in the language of this present paper, three sets of coplanar 

 axes, with one common axis, e. g. (127) = (457) = (367) = 0. If there are 

 four sets of coplanar axes the equation can be integrated by quadratures. On 

 the other hand, if we have three coplanar sets, but not with one common axis, 

 e. g. (127) = (457) = (134) = 0, no general solution of the equation has been 

 obtained. 



Another application of the ideas developed in the text is to point transforma- 

 tions. If we regard Xi, Xi, Xz, of (9) as plane homogeneous coordinates, (31) 

 gives the most general quadratic transformation having three singular points 

 i8i, 182, Pi, and four fixed points /34, /Ss, /^e, 0i- 



