891] 
9 
891. 
ON THE BINOD AL QUAKTIC AND THE GRAPHICAL REPRE 
SENTATION OF THE ELLIPTIC FUNCTIONS. 
[From the Transactions of the Cambridge Philosophical Society, vol. xiv. (1889), 
pp. 484—494. Read May 6, 1889.] 
I approach the subject from the question of the graphical representation of the 
elliptic functions : assuming as usual that the modulus is real, positive, and less than 
unity, and to fix the ideas considering the function sn (but the like considerations 
are applicable to the functions cn and dn), then the equation x + iy = sn (x' + iy) 
establishes a (1, 1) correspondence between the xy infinite quarter plane, and the 
x'y' rectangle (sides K and K') : viz. to any given point x + iy, x and y each positive, 
there corresponds a single point x' + iy', x, y' each positive and less than K, K' 
respectively : and conversely to any such point x' + iy', there corresponds a single point 
x + iy, x and y each positive. 
I draw in the x'y'-figure the rectangle A'B'C'D' (sides K and K'), and in the 
xy-figure, I take on the axis of x, the points B, C where AB = 1, = and the 
point D at infinity. 
C. XIII. 
We have thus in the x'y'-figure the closed curve or contour 
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