:344 HITCHCOCK. 



Let this value of 7, and the expanded form of Fp from (31) be substi- 

 tuted in (40). By equating to zero the coefficients of 

 •T?, .Ti.To, etc. we obtain six equations, 



pioi = 0, pi.4ii + ^2(03 + .-loi) + 2h{ao. + ^31) = 



^202 = 0, jJiUis + Av2) + i>^422 + pziai + ^32) = (42) 



Psaz = 0, 2J] (ao + -4i3) + i>2(fli + .403) + JJ3.433 = 



In order that 7 may exist, not all the p's can vanish. We shall then 

 have three cases, according as three, two, or one, of the as shall be 

 ^ero. 



Case 1°. «! = flo = as = 0. Values can be found for the p^s if, and 

 only if, the determinant of the ^'s vanishes. The vectors a of the 

 right side of (2) are then coplanar, and three of the four axes 184, 185, 

 ^e, 187 lie in this plane. 



Case 2°. Two only of the «'s are zero. Suppose ao = as = 0, ^^•ith 

 Gi not zero. Then 2h = 0. The six linear equations reduce to 



P2A21 + psAzi = (43) 



2^2^22 + P3(ai + ^32) = (44) 



P2(ai+ ^23) + PsA 33 = (45) 



INow p2 and ps cannot both vanish if 7 exists. Hence 



01.4.21 + -421.432 — ^31^22 = (46) 



01.431 + ^31-423 — .421.433 = (47) 



0? + ai(.423 + .432) + ^32.423 - .422.433 = (48) 



"From (46) and (47) we have 



0, Aoi, A31 



- .421, ^22, ^32 = (49) 



^31. .493, .433 



Tf A21 and ^31 are not both zero, oi can be found from (46) or from 

 (47). .\lso if Aoi and ^31 are not both zero (48) is a consequence of 

 (46) and (47). Since pi is zero, the plane of the coplanar axes con- 

 tains |8i. 



If A21 and ^31 are both zero, two values of oi are found from (48), 

 in general distinct, and there are two coplanar sets ^x^t\\ /3i in common. 



In any case, if the determinant (49), or either of the similar de- 

 terminants obtained from it by advancing subscripts, is zero, there 

 is at least one set of coplanar axes. 



