132 RESOLUTION OF SOLVABLE EQUATIONS 
§7. By (5), because uj us = Ug uzg,a = 0. Thus, one of the six 
unknown quantities is determined, while we have still the six equations. 
(15) to work with. It might be sufficient to say, that, from six 
equations five unknown rational quantities can be found. I will 
recur to this idea; but in the meantime the following line of reasoning 
Biante 
may be pursued. From (11), A = mo, Therefore equation 
(12) becomes 
gps = — 20 (k® — c? z) + 59%. (16); 
Also, because a = 0, equations (7) being kept in view, 
1 
w= — 
1 @ 
2 (K2 — o22) (b+ eg/2)—g3 (lk — ev/2) + 2k-+-04/2) (0° —9%e) hea/z? 
+ BY + BY’ /2) Js. 
+. Bo? =k } 2 (ke — c2 x) — g® } + Qchez (8? — ¢® 2) 
and By =c i 2(e—e2z) + 9 \ 4. Okhe (62 — ¢? 2); 
us “[k (248 — cz) — 9 § + Aches (6 — 9% 2)] 
‘ 
t 
+ <= [e }2 (k2 — cz) + g® | + Qhhe (02? — g?z)] + Vs. (17) 
Substitute in the second of equations (15) the value of B that has 
been obtained. Then 
g? ps = — 4k }2 (kK? — cz) — g a 8 chez (9? — g*z). (18) . 
The values of B’ and B’” are 
BY eg? = 0 {M + 2 ( — cz) } — g2zN = eg’, 
B" eg? = 9 N — 9 | M — 2 (hc? 2)) = 0; (19) 
where M = — 2 (A? 4 c? 2) + g3 + 4 kez, >) 
which may be written IJ = dkcez — P, 
20), 
and V = 2 (k? + c?z) — g§ — 4 ke, (2°) 
which may be written V = P — 4 ke. J 
The two equations (19) give us 
O{ M2 — 2 N® 402 (k2 —c2 2)®) = og? {Mf — 2e (R2 — ca), 
gp} M? — z N2 —4e? (k? —c*2)?} = eg? WV. | 
Therefore r (21) 
M — 2e (k? — c? 2) | 
J 
uae. 
2. = N 
