378 HITCHCOCK. 



This determinant is not zero, i. e., by hypothesis j3i, ^2, ^3 are not 

 coplanar. That these /3's are axes of F{p) is equivalent to writing 



FO81) = ci^i; F(/32) = C2^2; F{^z) = c,^z (7) 



Ci, C2, C3 being constants; whence (4) and (5) yield 



Ci + ybn + qbv2 + r6i3 = 



C2 + 23&21 + ?&22 + ?'&23 =0 (8) 



C3 + 2^&3i + 9^32 + rbss = 



three linear equations in the three unknowns p, q, r. Since (123) 

 does not vanish, the solution is uniquely possible. Let the values of 

 p, q, r thus determined be jpQ, qo, ro, and write 



Foip) = Fip) + top 



The relations (5) are then equivalent to 



Foifii) = 0, i^o(^2) = 0, Foifiz) = 0. 



The vector Foip) is most simply expressed in a new coordinate system 

 given by writing p = ^1X1 + ^2, X2 + ^3, .T3, equivalent to 



(23p) ^ (31p) (12p) 



"^=(m)' '^^=0^' -"^=(1^' ^^^ 



where (23p) denotes the determinant * 



with similar meaning for (31p) and 12p). If we write j8i for p we have 

 simultaneously a-2 = and xz = 0. Since i^o(i8i) vanishes in all its 

 components, no component can, in the new coordinate system, con- 

 tain any term in xi^. Similarly, no component can contain any term 



4 Using scalar products of three vectors, we may write Xi = r,a o o ^'^^ 

 similarly for X2 and X3. 



