OF THE FIFTH DEGREE. 133 
0? + g?z+ 209 
Equating the value of obtained from (21) with 
+ gz + 20¢2 
that derived from the last two of equations (15), 
2 Ee 
2khez 
ju — 26 (ih? — co? z) 1}? + N22 + 2 2N} M — 2¢e (kK? —c? z )} 
4M — 2e (ki — cP z)h? + + N%2+ 2Nz a} MU — 2e (h? — P2z)h aa 
The coefficients pz, p3 , etc., in the equation F (~) = 0, being given, 
g and & are known by (6) and (10). Therefore, by (16), c? z is known. 
Then (22) will be found to be a quadratic equation determinative of c. 
For, keeping in view the value of P in (20), (22) may be written 
M+ ez— ge 
2ke* z 
{4 (A? + c2 2)? + P? { — 8kPc — 16k (kK? — c? 2) (ce) 
14 (kK? — ¢? z)* — 16K? 2? z— P? c+ 8ke? 2P—4 (k? — c? z) P (ce) 
Because g, k, c? z and P are known, this equation is of the form 
H (ce) = Ke + L, 
where H, K, Z, are known. Therefore, since c? e? = c?z — c’, 
ce? (H?2 + K?) + 2KIc + (2 — H?2 22) = 0; 
from which c is known. Therefore, since c? z is known, z is known. 
Therefore e is known. Therefore, by (21), @ and ¢g are known. 
Therefore, by (18) or either of the equations (9), A is known. There- 
fore, by (17), u? is known. In like manner, wu}, uw}, wi, are known. 
3° 4? 
Hence finally, by (3), 7 is known. 
§8. Hxample First. I will now give some numerical verifications 
of the theory. The Gaussian equation of the fifth degree for the 
reduction of z!! — 1 = 0, when deprived of its second term, is 
22 11 ll x 42 bl ~% 88 
i Sy ne a ees a  (), 
i 25 |) 125 ame BL 
When a root of this equation is expressed as in (1), the value of 7, 
as given by Lagrange, is 
pe 
7 oo 
which, reduced to the form that we have adopted, is 
2189) 725, ./5 + 5 (19a a \(— 5 Dea 
