336 HITCHCOCK. 



^21 =+ (314) (561) (236) (167)CiC6+ (316) (451) (234) (417)CiC4 (25) 



For ^31 in a precisely similar manner, 

 ^31 =+ (124) (561) (236) (lQ7)CrC,-{- (126) (451) (234) (417)CiC4 (26) 



It is evident that expressions of similar form might have been found 

 by eliminating either C4, or Ce instead of C5. 



The other six ^'s may be found from (21), (25), and (26) by advanc- 

 ing the numbers 1, 2, 3, cyclically. 



It is also evident that, by means of identities of the same form as (23) 

 and (24), which connect any three C's with one another, we may 

 express any one of the A's in terms of any pair of C's we wish, the 

 coefficients being composed of scalar products of three axes. 



5. Application to Cases of Coplanarity of Axes. 



As a first illustration of the utility of the A's, we may inquire what 

 simplification takes place in the form of a quadratic vector when a set 

 of three axes become coplanar, — a question intimately related to the 

 problem of finding particular solutions of differential equations.* 



One obvious answer is that if the three coplanar axes be numbered 

 4, 5, and 7, the general expression (5) loses its third term and becomes 

 a binomial vector; but the form (5) implies a knowledge of all the 

 axes, while the data of the problem frequently furnish only three 

 axes, — for example, quadratic point transformations are often 

 specified in terms of the singular points, which are our three axes 

 j8i, 32, (Ss such that F^ is null. Finding the remaining four axes and 

 constructing the general expression (5) would then require, in general, 

 the solution of an equation of the fourth degree. I propose to show 

 that, if we have the quadratic vector in the form (1), or its equivalent 

 (2), it is never necessary to find the other four axes in order to detect 

 the existence of coplanarity among the axes; and, conversely, we may, 

 by giving proper values to the ^'s, immediately construct a ciuadratic 

 vector possessing any recjuired relations of coplanarity among the axes. 



Taking the latter problem first, as being the simpler, if it be merely 

 required that one set of axes be coplanar, we may let either of the three 

 scalars An, A-22, or ^133 be zero. This is at once evident from the form 

 of (21). For example, if An is zero, one of the four axes 1S4, 0$, ^%, 3i 

 must lie in the plane 3i, 3s- By a proper choice of the vector 8 in (1), 



4 See note to C. Q. V. page 385. 



