QUADRATIC VECTORS. 401 



as a normal form for a quadratic vector with two double axes, 185 and 

 iSe, which may be renumbered at our convenience. The vector a is here 

 the line of intersection of the tangent planes to (3) at the double axes. 



The symmetrical form (90) is possible only when the three similar 

 axes are not in the same plane. If they are coplanar, some of the 

 methods previously described may be used instead, e. g., the function 

 Q may be used.-^^ 



For three double axes we might write Q for P in this last result, it 

 l)eing now necessary to take, in general, a distinct from the a of (89). 

 For a symmetrical formula, however, we shall best return to (61), 

 and impose upon the nine A's conditions that /3i, 182, 183 shall all be 

 double axes, viz. that the determinant (63) shall vanish together with 

 two others obtained by advancing subscripts. If we wish the single 

 axis to appear explicitly, we shall most easily begin with the general 

 normal form (31), writing /«i/34 + wi/3i instead of /34, m2iS5 + 712182 

 instead of /Ss, and viz^e + "3183 instead of /Se. As lUi, m^, and vis ap- 

 proach zero we have the limit 



^i(237 ) (P25P36PP) , /32(317) ( PsgPhPp) , ^3(1 27) {PuP2,Pp ) .q^. 

 (123) (P25P36P7) (123) (P36P14P7) (123) (P14P26P7) ' ^— 



as a normal form for a quadratic vector having /Si, 182, Ps, for double 

 axes, 187 for a single axis, and the planes (14p) = 0, (25p) = 0, and 



13 As a simple example leading to a vector of the type (91), let it be required 

 to investigate whether the equations 



dx fly dz 

 xy yz zx 



can be integrated by quadratures. The integration depends upon that of 

 the i)artial differential equation SFp^u = where 



Fp = ixy + jyz + kzx 

 and hence upon 



."ipdpFp = 0. 



We easily find that i, j, and k arc double axes of Fp and that i -\- j -\- k m the 

 single axis. The tangent planes to (3) at i, j, and k are found liy taking polars 

 of VpFp thus,— 



Sp'v-VpFp = Vp'Fp + Vp[i{x'y + xy') +j{y'z + yz') + k{z'x + zx')] = 0, 



which, putting p = i,y — z = 0, x = 1, gives the single scalar equation 2' = 0. 

 Similarly we have x' = and y'= atj and k, respectively. Thus the vector 

 Fp is a limiting form of a type having three sets of coplanar axes, in the three 

 coordinate planes, the sets not possessing an axis common to all. Hence the 

 equation is not reducible to quadratures by any known process, (Cf. note to 

 Art. 6), nor even to a Riccati equation. 



