350 



HITCHCOCK. 



Finding the axes of F2P in a similar manner we have the matrix 



whence the relations of coplanarity (126) = (234) = (315) = (147) 

 = (257) = (367) = 0, so that F-ip belongs to the same type as Fip, with 

 central axes /3i, 182, ^z, and ^^. 



The vector Fzp is of different type. The plane .T3 = contains 

 only the two axes /3i and /So. Besides /Ss we have the axes (0, ^22, ^23) 

 and {hn, 0, 631) together with two imaginary axes in the plane 



h-2'2!>Z\X\ = &23^n.T2 



where x^ and 0:3 satisfy the quadratic 



622.T3" — &23&22^2^3 + 2bo3^X2^ = 0. 



We see that Fp is not necessarily of so restricted a type as Fip. 



13. Case w^here Three Given Axf.s form a Rectangular 



System. 



Of special interest is the case where the three assigned axes |Si, ^2, ^3 

 are mutually perpendicular both because the four solutions (59) are 

 no longer linearly independent, and l^ecause we might suspect here 

 some greater analogy with linear vector functions. We now have 

 bn = b'lz = bzi — and the equations (58) have the evident solutions, 

 (letting /, j, k replace j5i, ^2, l^z, so that ^u = ^22 = ^33 = 1), 



Fip = ///:: + jz.v + k-ry 

 Fip = jxy + kxz 

 Fsp = kyz + /.r.(/ 

 Fip = ixz + jyz 



which are linearly independent. The last three solutions are, to be 

 siu-e, reducible, but linear comliinations of them will not in general 

 be so. Thus in all cases the general solution may be expressed as the- 

 sum of four simpler solutions. 



