412 HITCHCOCK. 



any vector perpendicular to VSpFpdp, (i. e. dp is any vector tangent 

 to the cones (3)). The vanishing is independent of the value of dp, 

 corresponding to the arbitrary constants of (130). 



The reader familiar with Hamilton's theory of linear vectors will 

 perceive the condition for a double axis to be identical with the condi- 

 tion that the vector function VSSppFp, linear in 8p, shall have two roots 

 of its symbolic cubic equal to zero. 



21. To obtain a normal form for a quadratic vector having ^i 

 as a triple axis, we may apply the method of limits to (31). Let 

 1S4 be replaced by mjSi + ?7/34 and let w approach zero. In the limit 

 we obtain, somewhat after the manner of (91), 



fe(567) iP,P,P p) h{617) {P,PuPp) , . h{ 157) jPuPtPp) 



^' (156) (P5P6P7) (156) (PePi.P^) (156) (P14P5P7) 



(135) 



This vector has |Si for a double axis, with the tangent plane to the 

 cones (3) that of /3i and ^i. We may now cause /St to approach /Si as 

 a limit by writing, instead of jSy, the vector /Si + x^i + cx^^^, and at 

 the same time putting hx for h. The constant c is arbitrary except 

 as noted below. When .r approaches zero, the vector (135) approaches 

 the limit 



. h(P,P,Pp) , ^ h{6U) {P,PuPp) 



(P5P0P14) (156) (P6Pi4[P4 + cP,^]) 



A(154) jPuP^Pp) ..... 



+ ^' ri56HPHP5[m^7]' ^ ^^^ 



which may be taken as the required normal form. With regard to 

 the denominators of the second and third terms, the cone 



(PPP14P4) + ciPpPuPii) = (137) 



passes through the axes /3i, 1S2, and jSs, with tangent plane at |8i the 

 plane of jSi. and /34, and with principal curvature determined by the 

 constant c. If the left of (137) be abbreviated Dp, the denominators 

 of the terms in question become, respectively, (156) De and — (156)Z)5. 

 These denominators do not vanish, therefore, so long as the cone 

 (137) does not contain as elements either /Ss or /Se. But this cone is 



