( 466 ) 
Eliminating alternately the ws and 4’s from the five last equations 
(15) we have ' 
Bb ae bbl eha) 
6b,4,? — 6b,4,A, + 26,4,? + b, (84,4, — AA) = 0 | 
(Bu, A (U4) b, sE 2u, 'b, a 6u,b, =f 6d, = 0 
ub, 8 (Bu, a uu) b, = 6u,5, a 36, = y 
(16) 
The two latter equations (16) enable us to express u, and mw, in 
function of u,. For multiplying the first of these by 25,, the second 
by 4,, and adding up, we find the following quadratic equation 
(u,b, — 2u,6,)? + 6d, (u,b, — 2u,6,) + 3 (40,5, — 5,6,) = 0 
so 
u,b, — 2u,b, = — 3b, + V8, 
where the square root stands for both values, and 2, represents the 
expression 
i, = 3b,? — 4b,b, + bb 
This result, in connexion with 
u, (4,5, a 2u,b, + 65.) = 36, = bu), 
gives 
ant 3b, st bu), ris 6b, a Sub, 
REVE ae 
Now the first seven equations (15) lead up to 
Hy 
a 2).0 0 0 0 SOR iO a DT 6 OD 
gee 0e Dar AD " En MOE Oak OE 
a, Ui u, l 0 =d, —A, u, u, 1 0 —Ay —A, 0 
| i eee 
a, =o a, Wo u, u, 1 —A, ay >= Uy U Wz 1 
ja 0 De Pa fh, — 4% —A, 0 Lo Lh, fly eG oe —Ag| 
0 Wo u, 0 —A, 0 0 U, u, 0 aks 4; 
lo 0 0 O pp 0 Of HO 0:0 pp, OO 
which reduces to an equation of Rrccarr. For adding up in the 
numerator 4, times the third column to the sixth and 4, times the 
fourth to the seventh, and in the denominator 2, times the second, 
the third, and the fourth columns respectively to the fifth, sixth, and 
seventh, we find 
