Hitchcock — On the Formulation of two Linear Vector Functions. 5 



we shall find the scalar products Spfp and Spdf> to he proportional to the 

 quadratic expressions 



rx 2 + 2pxg 4- qz* and cp.f + Ciqz 2 ; 



If we now combine the two equations so as to eliminate ~ 2 from the former, 

 we may obtain the terms of the second degree as 



ax z + 2pxy and p^x* + q^, 



where a, />,, and q y are constants. The form of the result shows that the curve 

 has a double point at infinity. 



5. Typical form of <p and 9 when (j>- x 9 has a triple axis. — Suppose <j>~'9 to 

 have a triple root. The most general form of <fr l 9 is* 



(j)^9p = gp + cfiSfiyp + C-iySyap, (8) 



where g is the triple root, and may vanish. It is assumed that c,c lt and <Sa/3y 

 are all different from zero. As before, we regard a, j3, and y as known, 

 operate on them by <p, and call the results A, p., and v. Whence 



(pp . Saj3y = XSjiyp + pSyap + vSafip, (9) 



and by operating with <j> on both sides of (8), 



dp = g(pp + CpSfiyp + C,vSyap. (iti) 



This is, perhaps, the most convenient form of 9, but we may, if we wish, bring 

 out the analogy with (1) and (6) by writing 



c^cSafiy, c, = crafty, ajSapy =V0y, j3 3 -S'aj3 7 = V$y, and y3 &i|3y = Fa/3, 



when we shall obtain 



(j>p = \Sa 3 p + pSj3 3 p + vSy 3 p, ~\ 



Op = (gX + c 2 p)Sa 3 p + (gp + c 3 v)S(5 3 p + gvSy,p. J (10) 



It is clear that 9 differs from (1) in its effect on a and on j3. The only 

 direction similarly altered by <f> and 9 is y, the triple axis of <p~'9. 



6. Restriction to self-conjugate functions. — If, as in Art. 4, we require that 

 <f> and 9 shall be self -conjugate, we have by (9) and (9^ 



V\ V$y + Vp Vya + VvVafi = 0; Vcp F/3 7 + Vc,v Vya = 0, 



which, c and c, not being zero, are equivalent to 



Sj3» = 0, Syv = 0, Syp = 0, Sav = Sy\, S(3\ = Sap, and cS(Sp = c l Sav. 



Solving for A, p, and v we have 



X =?• V(5y + q Vyu + cp Vafi, n = q Vfiy + c x p Vyu, v = op Vfiy, 



* Loc. oil., p. 17-t, equation (24). 



