338 HITCHCOCK. 



2°. We may let A22 and As^ vanish and let A23 = A^i. The three 

 sets of coplanar axes have iSx in common. 



The differential equations corresponding to these two cases are of 

 very unlike character. In 1° the existence of tjie coplanar sets is 

 evident from the form of the ^'s. To prove it for 2° we may let the 

 vector 5 of (1) be expanded thus, 



5Sj8,/32^3 = aim/33 + (hVfi^fi, + a3r^^i^2 (28) 



and since we have identically 



p = ^iXi + iS^xs + ^3x3 (29) 



we shall have 



S5p = ai.Ti + a2.T2 + 03.^3 (30) 



By expanding Vippdp as in (2), using the values of the a's from (4), 

 and Sbp from (30), the fundamental equation (1) may be expanded 

 in the frecjuently useful form 



Fp= /3i[ AiiX'i^z -^ {(h-\-Aii)XzXi-]-{ai.-\- Aiz)XiX2.-\- a\x\\ 



+ /32[(a3 -\- A-i^XiXz + Ai^iXzXi +(ai+ ^23).ti.T2 + a2X^2[ C31) 



+ /33[(a2 +^31)^23-3 + (oi +^32)a;3a-i + ^33a'iX2 + Czxl] 



This is a way of expressing a quadratic vector which is always possible 

 except in the very special cases examined in C. Q. V. where a set of 

 three diplanar axes cannot be found; this expression depends on the 

 twelve constants which occur explicitly, and on the three directions 

 ^u ^2, ^i of the diplanar axes, equivalent to six more scalars. If now 

 in addition to the three conditions ^22=^33=0, ylo^^ylso just 

 assumed, we take 



tti = —As2, ci'i = 0, as = 

 the quadratic vector takes the form 



Fp = !Si[^ua;2a-3 +^i2X3a;i +Ai3XiX2 -^23^-?] + (182^21 + S3^3i)a-2.r3 (32) 



But this is a binomial; it must, therefore, by the reasoning of C. Q. V. 

 page 384, possess three axes in the plane of the vector coefficients jS] 

 and (182/121+ jS3^ 31) • But we already know it possesses three axes 

 in the planes /3i, 182 and 181, /Ss, respectively. It therefore has three 

 coplanar sets with j8i in common. For brevity an axis common to- 

 three coplanar sets may be called a central axis. 



