§06 MR. A. R. FORSYTH ON THE THETA-ITjRCTIORS, 
By —AS =1 
A'S'-By=l 
By -A'S=0 
By' - AS' =0 
hence 
B_A_1 
S' 7' 78'—7'8 
B_'_A'_1_ 
8 7 78'—7 '8 
and therefore A, B, A', B' are determinate functions of k 1} k 2 , k 3 . The equations (58) 
are the well-known equations for the hyperelliptic integrals of the first kind. 
On the expressions of the quarter-periods as definite integrals. 
18. Integrating (58) 
= 1 :^+ 1 : 
=-dx 
plA'+B'# 7 , p*A'-fBA 7 
y=\»^sir dx+ \ i ^r dx 
, 0 y/X ~~ 1 J fl y/X 
fiA'+B'* 7 . p*A'+B'# 
a , b being constants : we proceed to find some integrals giving the values of the 
periods. 
19. (i.) Let £c= 0 , y= 0 , so that all the uneven functions vanish; then, by (52), 
£C 1 = 0 , x 2 =— ; and hence 
[V A + Bx 
°=f 
°=L yx 
v/X 
.“A' + B'.t- 
dx 
dx. 
(ii.) Let x=K, y=0 ; referring to formula (8) the functions which vanish are seen 
to be 3- lt $ 3 , $ 9 , & 10 , & 14 , & 1B ; and to ensure this aq=l, x 2 —\ so that 
Tr PA + Bar 7 , piA+Baj 7 
K =\ 0 ~vr dx H vr dx 
