A THEOREM IN ELLIPTIC FUNCTIONS. 
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integral u — u 0 = v — v 0 ; while the foregoing integral equation, on reducing the constant 
coefficients contained therein, takes the form 
— k' 2 sn u sn v sn u 0 sn v 0 
+ cn u cn v cn u 0 cn v 0 
— dn u dn v dn u 0 dn v 0 
_ k' 2 
~ k 2 ’ 
viz. this equation holds good if u — u 0 = v — v 0 . And by a change of signs we have 
the theorem. 
If, as above, w+v+r+s= 0, the theorem gives a linear relation between the 
three products sn u sn v sn r sn s, cn u cn v cn r cn s, dn u dn v dn r dn s, and regarding at 
pleasure the sn’s, the cn’s, or the dn’s as rational, one of these products will be 
rational while the other two will be each of them a quadric radical; and hence, 
rationalising, we obtain an equation which contains the product in question linearly, 
and contains besides only the squares of the sn’s, cn’s, or dn’s; that is, we have 
three such equations containing the three products respectively. Bringing to one side 
the terms which contain the product, and again squaring, we obtain an equation 
involving only the squares of the sn’s, cn’s, or dn’s; but the three equations thus 
obtained represent, it is clear, one and the same rational equation, which may be 
expressed as an equation between the squares of the sn’s, or of the cn’s, or of the 
dn’s, at pleasure. This equation may be obtained, as I will show, from the ordinary 
addition-equations of the elliptic functions, but it is not obvious how to obtain from 
them the three equations involving the products respectively, and these last have the 
advantage of being of a degree which is the half of the equation which involves 
only the squared functions. 
Write x, y, z, w for suit, sn v, snr, sns respectively; then, writing 
we have 
that is, 
and consequently 
A =£cVl — y 2 .1 — k 2 y 2 , 
A' = y Vl — x 2 . 1 — k 2 x 2 , 
P — x 2 — y 2 , 
D = 1 — k 2 x 2 y 2 , 
sn (u + v) = 
A +A/ 
D A-A'~~ 
a = z V1 —w\ 1 — k 2 w 2 
a = w Vl — z 2 .1 — k 2 z 2 , 
xz = z 2 — w 2 , 
8 = 1 — k 2 z 2 w 2 , 
— sn (r + s), 
a + a' _ -nr 
8 a^a' ’ 
whence 
Dot = - (a — a') (A + A'), 
P8 = — (a + a') {A - A') ; 
DOT-P8=2(Aa'-A / a),