0«=*=^, &c. 



0/ Einstein's Gravitation Theory, 235 



we find, by (5), 



}-.... (6) 



J 



These values of XI.. are to be introduced into the cubic (4), 

 giving the three roots a i} cr 2 , <r 3 , of which two will, no doubt, 

 be equal. These, however, can be found more simply by 

 remembering that the cubic is precisely of the form of the 

 equation for the principal axes of the linear vector operator m 

 whose constituents are X2 n , Q 12 = fl 2 i ? &c. Call P the operand, 

 denote otP by P', and use the suffixes 1, 2, 3 for the com- 

 ponents along the axes of a ln # 2 , #3. Then, by (6), and 

 putting for the moment F 2 -^F/r = '9' 9 



p/=^ 1+ 5*gp 1+ ^p 2+ ^p s ), 



or, calling u the unit of r, 



P/ = ^P 1+ ^(|>(Pu), 



with two similar expressions for P 3 ', P 5 . Thus, for any 

 vector operand P, 



«xP=¥-.P + $.u(uP). 



Whence we see at once that the principal axes are radial 

 and all transversals. In other words, the operator is radial!// 

 symmetric. To find the principal values, i. e. a, take first 

 P-Lu, and then P || u, that is, radial, obtaining in the first 

 case 



■bxP=¥P, i. e. o- 1 = o- 2 =^ r , 



and in the second case, 



«rP = (^ + 0))P, i. e. cr 8 = ¥+$. 



Thus, remembering the meanings of V F and<3>, the required 

 roots of the cubic (4) will be 



<r 1 — a. 2 = F 2 + - F (transversal) 1 



and *«, Y • • • ( 7 ) 



0-3= d y--~F 2 (radial) ! 



dr ' j 



The corresponding principal curvatures will be, by (3), 



K i z=^ K p + a l , i- 1, 2, 3. 



