378 



HITCHCOCK. 



This determinant is not zero, i. e., by hypothesis jSi, ^2, /Ss are not 

 coplanar. That these jS's are axes of F(p) is equivalent to writing 



F(^i) = ci^i; F(/32) = C.M2; F(^z) = cz^z 

 Ci, Co, C3 being constants ; whence (4) and (5) j'ield 



Ci + pbn + 9^12 + rbiz = 

 C2 + phi + 9&22 + r&23 = 



cz + pbzi + qh2 + rhsz = 



(7) 



(8) 



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 po, qo, ro, and write 



Foip) = F{p) + top 



The relations (5) are then equivalent to 



Foifii) = 0, Fom = 0, Fo(/33) = 0. 



The vector i^o(p) is most simply expressed in a new coordinate system 

 given by writing p = /3i .Ti + ^2, X2 + 183, .T3, equivalent to 



Xi = 



{23p) 



X2 = 



(31p) 



(123)' " (123) 

 where (23p) denotes the determinant * 



^3 = 



(12p) 

 (123)' 



(9) 



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

 simultaneously X2 = and xs = 0. Since Foifii) vanishes in all its 

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

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



4 Using scalar products of three vectors, we may write Xi = ^„ ^ ^ and 



similarly for X2 and X3. 



5/?l/?2/93 



