THE AXES OF A QUADRATIC VECTOR. 341 



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



wliich 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 .rj = A'i(a'i, ^-z, a's), 4 = X^^xi, x^, Xz), xl = X^ixi, xi, x^), 

 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 



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



where 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 which occur in the form 

 of a quadratic vector when sets of coplanar axes exist, let it be required 

 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, Aoji, .I33, and Oi, ao, 03, all vanish. The 

 cjuadratic vector then takes the form 



Fp = /3i(/li2.T3a-i + A13X1X2) + ^2(-'l21.T2.T3 + AosXiXo) + 



i33(.43i.r2.i-3 + .I32.v3.r1) ; (33) 

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



/34 = iSo.'i.i + /33.I31; 135 = i33.l32 + Mn; 13, = Mu + M23. (34) 



To find iSy we note that, by C. Q. V. page 385, this axis is perjDendicu- 

 lar to the three values of 8 by which we may pass from one binomial 

 form of Fp to another. Thus by an easy calculation 



,3- = S f/3i(.4i2yl3i + /l2i.4i3 - .421.431)] (35) 



123 



