454 HITCHCOCK. 



Summary of Results. 



38. All irreducible quadratic vectors may be thrown into the 

 form Vcppdp + pS8p, with exception of two special cases. 

 First exception. 



^3(ax,X2 + gx^'' + g.x^xz) + Mi + b3^z)xi'. (241) 



This vector may be detected, when its components X, Y, Z, are 

 given in any form, by the following properties; it has a sextuple axis, 

 and a single axis; the tangent plane to the cones (3) is unique and 

 determinate for at least one of the cones; the single axis lies in this 

 tangent plane; on taking /3i for the multiple axis, ^3 for the single 

 axis, (so that av = gives the tangent plane), and ^2 any vector 

 without the tangent plane, we may remove the terms in xi^, a: 10:2, and 

 0:10:3 from the coefficient of /3i, by adding a properly chosen term pS8p; 

 and the vector then takes the form (241). 

 Second exception. 



^libiXiXs + UX3) + 133(633:23^3 + axiX2). (251) 



The properties of this vector are : it has a quintuple axis ; the tangent 

 plane to at least one of the cones (3) is unique and determinate at this 

 axis, which is an inflectional element of the cone; on taking ^i for 

 the quintuple axis, jSi for another axis, 183 any vector in the tangent 

 plane except /3i, we may remove the terms xi^, XiXo, and xiiCs from the 

 coefficient of 0i; the vector then takes the form (251). 



If the quadratic vector does not come under either exceptional case, 

 the method of throwing it into the form Vcjipdp + pS8p depends upon 

 the configuration of the axes. The form of the constant vector 5 is 

 not, usually, unique, but may have thirty-five possibilities or fewer. 

 The values obtained in this paper, are, therefore, not the only ones 

 that could be given for the majority of cases. They have been selected 

 so that 8 should be rational, and as simple as possible in terms of 

 known axes. 



Massachusetts Institute of Technology. 

 Boston Mass., May, 1916, 



