THE AXES OF A QUADRATIC VECTOR. 341 



iax^ + j[a'x'~ + b'xy + (c'- b)yn + k[Z - byz] 



which proves the theorem. 



The connection between the theory of quadratic vectors and Dar- 

 boux's treatment of differential equations is now fully established. 

 To continue the study of the above differential equation would be 

 merely to repeat Darboux's work. 



To indicate a quite different application of the present theory we 

 may note the following, — 



Theorem 7. If a quadratic 1 : 1 point transformation be defined by 

 the equations .r| = A^(.Ti, X2, xs), xl = A'^Gri, .T2, a'3), xl = Xzixi, X2, X3), 

 and if the four fixed points of the transformation be situated as follows: 

 the singular points being A, B, C, two fixed points lie on a straight 

 line through A, the other two lie respectively on AB and AC; — the 

 transformation can be written in the form 



X 



= tx, ])' = Y{x, y,) + ty, z' = Z{x, y, z) 



^here t is a linear function of x, y, z. 



This theorem is, of course, an immediate consequence of theorems 

 2 and 3, stated in the language of point transformations. 



7. Vectors with Four Sets of Coplanar Axes. 



Continuing the study of the simplifications wliich occur in the form 

 of a quadratic vector when sets of coplanar axes exist, let it be recj[uired 

 to have four such sets. Here, again, we shall evidently have two cases, 

 according as we have a central axis or not. 



If there is to be no central axis, we may begin with case 1° of Art. 5, 

 letting the six quantities An, A22, Ass, and ai, ao, as, all vanish. The 

 quadratic' vector then takes the form 



Fp = ^liAuXsXl + ^13.Tia-2) + ^2(A2iX2Xs + A2SX1X2) + 



|83(^3ia;2a:3 + As^xsXi) ; (33) 

 it is evident that three of the axes may be taken as follows, — 



^4 = ^2.121 + iSsAsu /35 = i33.-l32 + Ml^', /Sfi = ^lAu + 02A2S. (34) 



To find (S; we note that, by C. Q. V. page 385, this axis is perpendicu- 

 lar to the three values of 5 by which we may pass from one binomial 

 form of Fp to another. Thus by an easy calculation 



/37 = 2 MAnAsi + AoiAn - A^iAsi)] (35) 



123 



