350 HITCHCOCK. 



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



^6 



&■> 



1 



1 



1 



bi2 bn 



bn 623 



&13 &23 



O12O13 012023 O23O3I 



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

 = (257) = (367) = 0, so that F2P belongs to the same type as Fip, with 

 central axes ^i, ^2, ^3, and /S;. 



The vector Fsp is of different type. The plane 0:3 = contains 

 only the two axes /3i and (82. Besides (83 we have the axes (0, 622, ^23} 

 and (611, 0, 631) together with two imaginary axes in the plane 



boobsiXi = b2sbnX2 



where X2 and xz satisfy the quadratic 



6223:3^ — &23&22a'2a-3 + 2b23^X2^ = 0. 



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



13. Case where Three Given Axes form a Rectangular 



System. 



Of special interest is the case where the three assigned axes jSi, /So, 13$- 

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

 no longer linearly independent, and because we might suspect here 

 some greater analogy with linear vector functions. We now have 

 &12 = &23 = ^31 = and the equations (58) have the evident solutions,, 

 (letting i, j, k replace jSi, ^2, ^3, so that bn = ^22 = ^33 = 1), 



Fip = iyz + jzx + hxy 

 Fop = jxy -j- kxz 

 Fzp = kyz + ixy 

 FiP =- ixz + jyz 



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

 sure, reducible, but linear combinations 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. 



