338 HITCHCOCK. 



2°. We may let 7^22 and ^33 vanish and let ^23 = ^32- The three 

 sets of coplanar axes have /3i in common. 



The differential equations corresponding to these two cases are of 

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

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

 vector 5 of (1) be expanded thus, 



5S/3j/32^3 = aiF/3.>/33 + a2T?3/3i + asV^,^. (28) 



and since we have identically 



p = /3i.ri + /3>.r2 + i33.T3 (29) 



we shall have 



S8p = ai.ri+ a2.r2+ 0,3X3 (30) 



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

 and S5p from (30), the fundamental equation (1) may be expanded 

 in the frequently useful form 



Fp = /3i[ ^n.T2.T3 + (fl3 -^An)x3Xi + (a2+ ^i3)a:i.T2 + aixl] 



+ |82[(a3 +^2i).T2.r3 + AizXsXi + (ai+ .423)a-i.T2 + azxl] (31) 



+ ^3[(a2 +-43i).r2.T3 + (oi -\-A3y)x3Xi + ^33.ri.r2 + ^30:31 



This is a way of expressing a quadratic vector which is alwaj^ 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 

 |3i, |32, jSs of the diplanar axes, equivalent to six more scalars. If now 

 in addition to the three conditions yl22=^33=0, ^23=^32 just 

 assumed, we take 



«! = —^-132, «2 =0, as = 



the quadratic vector takes the form 



Fp = ^i[AnX-iX3 -\-A12X3X1 +.4i3.Tia-2 -.423.r?J + (02A21 + fl3^3i)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 (i32/l2i+ /33^3i). But we already know it possesses three axes 

 in the planes j8i, 02 and /3i, 1S3, respectively. It therefore has three 

 coplanar sets with /Si in common. For brevity an axis common to 

 three coplanar sets may be called a central axi^. 



