88 REPORTS ON THE STATE OF SCIRENCR.—1913. 
Part I. Exuieric Functions. 
Report III. on Tables of the Elliptic Function. 
Ten new tables have been calculated, for which the ratio of the periods 
Kau 3 
Ry /2, 2/2, 2/3, 3, J? 
The square root of a rational number was chosen as the period ratio, 
so as to utilise the singular modulus of the elliptic function which arises 
in the theory of Complex Multiplication, and thence obtain an independent 
numerical check. 
The table of the period ratio K’/K = /3 and modular angle 6 = 15° 
has been printed already ; also of K/K’ = /3, 675°; and thence the 
table for K/K’ =2./3 or 3/3 was derived by the quadric or cubic 
transformation. 
The table for 
3/2, 3/3, 4, 5. 
r GY “4 = 1 2 
K =4K',=(¥5 =) 
was calculated by a quadric transformation of K=2K’, given in 
Report II., and it could have been calculated immediately from K = K’ 
by a quartic transformation; and K =5K’, sin 26= (2 sin 18°)! was 
calculated by a quintic transformation of K =K’. 
A sketch of the table for 
K = 7K’, sin 20 = (2 ae tL Ye Pe 
is submitted, obtained from K=K’ by a transformation of the seventh 
order, with a view of showing the shape of the curve for E (r), D (r), A (r) 
in a penultimate form, when the modular angle is undistinguishable from 
a right angle. 
Curves of the function E (r), D (7), A (7) are given in the figures to show 
the change of shape as the modular angle 6 increases from 0° to 90°. 
It will be observed that these curves are featureless for @ up to 15°, 
and even to 45°, showing that the elliptic function does not require tabula- 
tion for a modular angle much below 45°, as E(r), D(r), A(7) may be re- 
placed by a circular function formula within the limits of accuracy of the 
four significant figures required in a practical problem. 
But in the important cases which arise in physical applications of a 
modular angle in the last degree of the quadrant it will be noticed that 
the curve of a function preserves a definite character in a penultimate 
form, even when the modular angle is undistinguishable from a right 
angle, provided the period ratio is assigned. 
The tabulation must be abandoned here which takes the modular 
angle as a parameter of the function ; and the period ratio K/K’, or else 
Jacobi’s ¢ = exp(—7K’/K), must be adopted instead, as the parameter of 
a table. 
To ensure the accuracy of a transformation formula employed in the 
calculation of a table the check values were applied at the beginning and 
end, *=0 and 90; half-way, at bisection, where r=45; and then at tri- 
