[818 
818] NOTE ON HYPERELLIPTIC INTEGRALS OF THE FIRST ORDER. 99 
The given equations then are 
(A/3 - Bol + C8 - Dry = 0, A’P - B'ol' + C'8' - D'y' = 0, 
(43) j A/3' - B'ol + C8' - B'ry = 0, A'/3 - Bol' + C'8 - Dy' = 0, 
[Aol' - A'ol+ Cy' - C'y = ¡7T, B/3' - B'/3 + D8' - D'8 = \ir; 
and it is required to show that these lead to the relation 
(49) AC' — A'C + BD' - B'D = 0. 
From the first and fourth equations, and from the second and third equations of (43), 
we deduce 
(AC' - A'C) /3 + (Col' - C'ol) B + (Cy' - C'y) D = 0, 
(AC' - A'C) /3' + (Col' - C'ol) B’ + (Cy' - C'y) D' = 0 ; 
TEGRALS 
and again from the first and third equations, and from the second and fourth 
equations of (43), we deduce 
] 
(BD' — B'D) a + (D/3' - D'/3) A +(D8'-D'8)C =0, 
(BD' - B'D) cl' + (D/? - D'/3) A' + (D8' - D'8) C = 0. 
Functionen,” 
lin equations 
for greater 
These pairs of equations give respectively 
AC'-A'C : Col'-C'ol : Cy' - C'y = BD' - B'D : D/3' - D'/3 : - (B/3' - B'/3), 
and 
'•&J' 21 = 0, 
« *4 =0, 
'» = £tt ; 
AC'-A'C : Col' — C'ol : - (Aol' - A'a) = BD' - B'D : D/3'-IT $ : D8'- D'8; 
whence putting for shortness Aol' — A'a, B/3' — B'(3, Cy'— C'y, D8' — D'8 = a, b, c, d, we 
have 
AC'—A'C c a , 
BD' — B'D ~ b - d’ whence ab ~ cd - 
But the last two of the equations (43) are 
id the eight 
r ’s, and there 
only, which 
But taking 
ir values as 
a + c = \tt, b d = 5 
we have thus a + c = b + d, =b + ^, =^(a + c); or, since a + c, =\nr, is not =0, this 
gives b = c, whence also a = d, and we have 
AC'-A'C 
BD' — B'D ~ lf 
— (o.,Vx = 0 oi 
Abtiliennes,” 
that is, 
AC' -A'C+BD' -B'D = 0, 
the required equation. 
he equations 
Cambridge, 10th September, 1884. 
' r r 
12 J u 21 j "22 
, 7 » S'. 
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