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 

 ■^l, X\X2, etc. we obtain six equations, 



piai = 0, 291^11 + ^2(03 + A21) + ^3(02 + Azi) = 



2^202 = 0, 2^1(03 + ^12) + 2^2^22 + psiai + ^32) = (42) 



Pstts = 0, 2^1 («2 + ^13) + P2(«l + ^23) + 2^3^33 = 



In order that 7 may exist, not all the 2^'s can vanish. We shall then 

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

 zero. 



Case 1°. ai = 02 = 03 = 0. Values can be found for the 2j's if, and 

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

 right side of (2) are then coplanar, and three of the four axes ^i, ^5, 

 /Se, /S? lie in this plane. 



Case 2°. Two only of the ft's are zero. Suppose a^ = 03 = 0, with 

 ai not zero. Then jh = 0. The six linear equations reduce to 



V2A21 + 2^3^31 = (43) 



:P2^22 + 2^3 («i + ^32) = (44) 



P2(ai+ ^23) + 2^3^33 = (45) 

 Now 2^2 and pz cannot both vanish if 7 exists. Hence 



aiA21 + ^21^32 - ^31^22 = (46) 



aiAsi + A31A2Z - ^21^33 = (47) 



0? + ai(^23 + ^32) + ^32^23 - ^22^33 = (48) 



Trom (46) and (47) we have 



0, A21, A31 

 - A21, A22, Az2 = (49) 



A31, A2Z, Azz 



If ^21 and Az\ are not both zero, cii can be found from (46) or from 

 (47). Also if A21 and ^31 are not both zero (48) is a consequence of 

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

 tains jSi. 



If A2\ and ^131 are both zero, two values of ai are found from (48), 

 in general distinct, and there are two coplanar sets with |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. 



