348 HITCHCOCK. 



12. Interpretation of the Equations (58). 



It is manifest that the equations (58) are biHnear: and that the set 

 of six 6's are determined by the choice of three axes (3i, 02 bg, while- 

 the set of nine A's go^'ern the other four axes. I shall first show how 

 various simple solutions of these equations may be wTitten down,, 

 and shall then discuss the general solution as a linear function of these 

 simple solutions. 



One solution is seen by inspection to be: An, A03,, Azz, proportional 

 to 611, 622, and &33, with the other six .4's all zero. The resulting value 

 of Fp, which we may call F\p, is, by (31), 



Fip = /3i6ii.r2.r3 + 1S2fc22.V3.T1 + ^sh^XiXi (59i)i 



which, of course, may be multiplied by a scalar constant. It is easy 

 to check the result by operating with V, using the values of the &'s 

 from (57i), and showing that the expression is of the form Vp8. 



A second solution is equally obvious: let .423 + ^32, ^31 + ^13, and' 

 An + An be proportional to ^23, ^31 and &12 with ^u = 7^22 =-^33 — 

 and A22. = ^132, Az\ = An, An = .42i, giving the solution 



F2P = Q\{hviXz.Xi + 613.r1.r2) + i32(/>2i.r2.r3 + fc23.V1.r2) + i83(fc3i.r2.V3 + fc32.r3.r1)' 



(592) 



We note that this solution is null if 623, 631, and fci2 are all zero, that is, 

 if the vectors jSi, ^-i, and 0^ form a rectangular system. Let us suppose 

 for the moment that one of the fc's, as h^z, is not zero. 



These two solutions are linear in the fc's, and are not altered by 

 advancing subscripts. A third solution, is found by assigning arbi- 

 trarily An = ^^22 = ^^23 + ^^32 = and solving for the .4's in terms' 

 of the fc's. The result is 



-4.33 = —-hsihos, Azi = h'zshu, Au = —b^shn, An = fc22fcii, 



-421 ^^ b-yibw, Aiz =^ — b-yjozu Azi = ~rfc22fc3ij- 



which may be checked by substituting in (58). ^Yhence by (31) 



FzP — ^libnbn^'zXi — fc23fc11.r1.r2) + ^libiibn^-i^s — fc22fc31.r1.r2) 



+ i33(fc23fcii.r2.r3 + fc22fc3i.r3ri — 2fc31fc23.r1.r2) {o9z) 



quadratic and unsymmetrical in the fc's. 



A fourth solution may now be obtained by ad\ancing subscripts, as 

 also, of course, a fifth, which might be used in place of F2P in case 

 all fc's with unec^ual subscripts vanish, but a simpler treatment of this- 



