400 HITCHCOCK. 



scalar equatioi} when operated on by Sp'V, if p = jSs- The cones (3) 

 do not, in general, have a double line at /Ss. When, however, we 

 choose one axis of coordinates along J85, at least one of these cones 

 passes twice through the double axis. 



14. A fourth method for multiple axes, while in some respects less 

 convenient, in that the tangent plane to (3) at the double axis is less 

 explicitly contained in the result, brings the present discussion into 

 close relation to the theory of point transformations ; — - instead of 

 C23) take 



Qip) = i-(12p)2 + j(12p) (2a'p) + k{2a'p) (a'lp) (89) 



This vector evidently has |3i for an axis. By polarization, it is obvious 

 that it has ^2 for a double axis, with the plane (2a'p) = tangent to 

 the cones (3) or meeting them twice at ^2. If, therefore, we replace 

 P by Q in the normal form (31) we shall have /3i, ^i, jSs, jSe, and ^7 as 

 single axes, ^2 as a double axis, and (2a'p) = the tangent plane to 

 (3) at j32. Fm-thermore, the result will be the most general quadratic 

 vector satisfying these conditions, aside from an additive term pS8p, 

 for it can otherwise differ from (88) only in the numbering of the axes. 

 Any vector wdth a multiple axis differs from another with the same 

 multiple axis only in the direction of the tangent plane to (3) at that 

 axis, a multiplicative constant, and the term pS8p; provided the five 

 single axes also coincide. 



15. These methods for multiple axes may be employed simul- 

 taneously to obtain two, or three, distinct double axes. Thus to 

 form a vector having /3i, jSe, and jSy for single axes, ^2 and /Ss for double 

 axes, we have only to write Q instead of P in (88). The vector a may, 

 in (89), be the same vector as in (88), i. e., it may be the line of inter- 

 section of the tangent planes to the cones (3) at 182 and jSs, — with the 

 obvious restriction that this line does not itself coincide with an axis. 

 In general we maj', if we wish, take a and a' any two vectors such 

 that, with the five axes, no six vectors lie on a quadric cone. 



More symmetrically, let the vector have /3i, (82, ^3, for single axes, 

 and two other double axes. By writing 13- = //m + "(Se in (88) and 

 letting III approach zero we easily find 



a{P,P,Pa,) {PsPePa,) {P^P^Pp) + ^,{P,P,Pa,y (P^Pa^Pp) 



+ ^e{P,P^Poi,y {Pa,P,Pp), (90) 



