338 



15 — 10 6 



and witli this determination of the surds we get 

 fA-ybc o-fl/co" h-\-V~^ 



•'-^==l+^y=:16, 0_:=,=:l+y«=-9, -I-— , = 1 + «^zr. -5. 

 /— I/Jc g—Vca h—\/ab 



The sum of the three fractions is equal to 2, therefore the integral 

 is reducible. At the same time we have found 



^7 = 15, y«= — 10, «/?=-6, 

 or 



«* = 4, ii' = 9, 7' = 25, 



so that we have either 



« = 2, /i=: — 3, r = — 5, 



or 



«= — 2, /?=:3, 7=5. 



Two sets of values for the constants «, I}, y being admissible, we 

 infer that the given quadrics ipi, lï'j, i|>, allow us to us build up two 

 entirely distinct involutions J, and instead of a single reducible 

 integral of the given form, two such integrals are possible. This is 

 obviously in accordance with the known proposition that, as soon 

 as an integral belonging to an algebraic function of deficiency /> = 2 

 is reducible, that function possesses a second degenerate integral. 



I will take up the case 



« = 2. I? == — 8, y = — 5. 



Then we have from (14) and from (13) 



P, P. P, 



1~-1~1' 



« _ /i ^ ^1 _ ^1 

 4 — 10 "~ ~ 1 

 hence from (8) 



Si bj 5j 



■4tp,-8i|j, 8ip, + 9t|j,-7i|7, _8v^-27i^, + 25i|j, 



or 



ecu 



.i-" ~ {x \y~ \' 



X 1 X — 1 



