ES eee ee SC ee SS S—“‘<is 
ON THE TABULATION OF BESSEL AND OTHER FUNCTIONS. 89 
section, 7=30, 60; and at quinquesection, when possible, where r=18, 
36, 54, 72 ; in accordance with the formulas of Report I., pages 6, 7, which 
provide an algebraical numerical value to contrast with the number 
obtained by the expansion of a q series, or else by the formula of trans- 
formation. 
These check values can be assigned for a singular modulus, and are 
given as they arise in the calculation of a table, and entered at once on 
the sheet of numbers to serve as standard points of reference in the same 
way as the cyclotomic values in a table of the circular function. 
Thus in all cases we have the check values— 
E(0)=0, D(0)=1,  A(0)=0, 
H(90)=0,  D(9)=F5, A(90)=1; 
, 
eG 1b PONG 1 
Rs) — 5", Das)=(Sur) A(t) =(5“5) stan $(45)= ,=D(00). 
The trisection and quinquesection formulas are given on pages 6, 7, 
Report I., and at full length in the ‘ Phil. Trans.,’ 1904, pages 261, 264 ; 
and they can be quoted as required in the check of a table. 
Thus the new table for K’/K=/2, x= /2—1=sin 24°-47, required 
the q series formulas for a complete tabulation ; and it was checked at 
trisection by taking b= /3+ V2. 
The table for K/K’= v2, «’=/2-1, can be derived by the second 
quadric transformation, and checked at trisection by b= /3—/2. 
Another quadric transformation gave K/K’=2./2, and _ here 
b?=(./2+1) (2+. 3) for trisection. 
Any function, such as A(r), may be distinguished as regards the period 
ratio by writing it 
K 
A ( ; @) - 
and the formulas of the second quadric transformation are written 
ee ae Kone 
D/ 90, — 
(5) 
ok Mey k (x w) 
1 22 = 7 JA > [P77 
P(r 2x7) D(x ra) Ae as 
K 
A(2r, ) 
K ae K 
a0) (2r, K’) si Ge p(2r,) 
K K’ 
1 (r, 25,)= Tied eee 
A( x)=A(s ae) P(r) 
