Vol,. 7, 1921 MATHEMATICS: L. P. EISENHART 
333 
Now (10) reduces to 
4 8 
which is consistent with (18). Eliminating a from the first of (18) and 
(19), we get 
5 „„_35^' + 5^ 2 _ 21^ 4 = a 
4 6 4 5 64 5 3 
If we introduce dependent and independent variables t and 0 by 
4 f e . 'Odd 
8 = e \ e dx h t = e — , 
J dxi 
we get the equation 
dd\de)^ 
t±\ t*? ) + 7t 2 ~ + 13*^ + (2f + 5) (3* + 5) (t + 3) = 0. 
dd dd 
To the evident solutions of this equation, t = — 5/2, * = — 5/3, cor- 
respond the solutions: 
8 = 25 *x 8/5 , a = - *r 2/5 , jS = 6 x^ 5 , 7 = - b 2 xf - 25 x* t 
5 = 25^ 12/5 , «=-6*! 2/5 , 0-6* v Vy--g(* + 25)** 
where b is an arbitrary constant. For t = — 3, we find a = 0, which is 
excluded since we assume that 8 > 0 in accordance with the theory. 
If we put t dt/dd = y, and take y for dependent variable, the above 
equation may be replaced by 
y % + y { 7t + 13 ) + ( 2t + 5 )( 3 * + 5 )(* + 3 ) = °' 
When a solution of this equation is known, the corresponding functions 
a., (3, 7 can be found by quadratures. 
5. Making use of the formulas of Bianchi 6 we find the following ex- 
pressions for the principal curvatures of the spaces (7), (8), (11), (14) 
(l + -*)ga = ft % + g;, ^ (7*) 
-1 = ^ = ^=2^75; 
Ii = - ^(1 + V3), J? 2 = p, f 3 = ^(I-VI). (14*) 
The principal curvatures for & ± 1 in (12) can be obtained explicitly, 
but their forms are quite involved. 
