40 REPORTS ON THE STATE OF SCIENCE.—1912. 
For this purpose it is convenient to follow Abel and take F(9) 
or hK as the argument in the Table, instead of g as in Legendre’s 
Table IX., and to proceed in the tabulation with equal increments of 
F(9), or AK, or ” K; dividing K into 90 degrees, instead of taking equal 
90 
degree intervals in 4, as in Legendre’s Table IX. 
r 
90 
equal steps of 1 degree in the quadrant; and then ¢ is tabulated in the 
column adjacent, and y in the column adjacent to F(p) or (1 — h)K or 
(1 _ 4) Ki and in Jacobi’s notation ¢ is the amplitude function of AK, 
In this new arrangement 7 and AK = —- K, or F(¢), proceeds by 
and denoted by 
(6) ¢ = am hK, v = am(l — /)K. 
Further, in Gudermann’s abbreviation of Jacobi’s notation 
(7) sin ¢ = sn AK, cos ¢ = cn AK, Ag =dn hK; 
and Jacobi shows that, expressed by the function D(r), A(r), B(r), C(r), 
of the Table 
_1 AG _ Be) = gi tale 
(8) sn hK= 77 Duy cnhK= Dir)’ dn hK = J Dir)’ 
implying 
(9) A(0) = B(90)=0, A(90) = B(O) =1, 
D(0) = 6(90) = 1, D(90) = C(0) = sa 
instead of Jacobi’s @(WK) and H(AK). 
The function D(r) and A(r), defined in (4), is tabulated, qualified 
by a denominator, because D(r), A(r). . . can be expressed exactly by 
surds at the aliquot division values of the quadrant, such as 
(10) r= 45; 30, 60; 18, 36, 54, 72; 15, 75,223, ... 
just as the corresponding values in a Circular Function Table, to which 
the Elliptic Function degenerates when the modular angle 0=0; and 
then 
(11) A(r) =sin 7°, B(r) = cos'r°, D(7) = CO). 5 
An important check is introduced thereby on the accuracy of the 
numerical calculation ; these check values are mentioned as they arise, 
and the algebraical formulas were stated in the Report, 1911, the theory 
being given in ‘ Phil. Trans.,’ A, 1904. 
For the Second Elliptic Integral the function E (r) is tabulated, or 
gt = E(90 — r); and these are given in terms of Jacobi’s Zeta function 
f 
(12) E(r)=anhK, F(r)=m(l —h)K; 
and connected with Legendre’s function E(¢) by 
E 
(13) zmhK =K(¢) — hE = E(9) — K Fi). 
