ON THE CALCULATION OF MATHEMATICAL TABLES. 53 



Also 



E(22|) + F(22^)=^4°4*-H , E(22i)-F(22i)=^M«^li5). 



Thus, for example, we find for 



K=2K', K' = (y2-1)-, a=^7r. 



8 



For the lemniscate function 



tan a=V2, =al 0-07525750=tan 49° 56'. 



But taking a=50° in a first approximation, with four-figure 

 logarithms, as a test of the formulas, 



sin(^7r— ia)=sin20 =al 1-5341 



sin ^a=sin 25 =al 1-6259 



sin (:^7r + a) = sin 95o=al 1-9983 



sin2a=sinlOO =al 1-9934 



sin (^+ ia)=sin 70°=al 1-9780 



cos a=sin 40° = al 1-8081 



Thus A(22i)^='^'''2^ ^^" 25(sm 80)* 



sin 85 

 =al 2-3200 

 A(22i)=al 1-5800=0-3802 

 R(99.i)4_ siP^70 sin^65(sin 80) ^ 

 sin 86 

 =al i-8601 

 B(22i)=al 1-9650=0-9226 

 j)/'22i)4_sin''65 sin 85(sin 80)* 



^' 2 sin270 sin'''40 



=al 00481 

 D(22i)=al 0-0120=1-026 



C(22i)* = sin^^n 85(sin 8 0) ' 

 2sin^20^n240 

 =al 0-2630 

 C(22i)=al 0-06575 = l-168. 



Testing r=22^ in Table II, where the angle a is exactly three quarters 

 of a right angle, 



31l7rll77rl 5 



=r' 4''-2"=16' r'^2*=16' r + ^^S'"' 



Ni 



a=r,7r. 



A/221 )4 ^ siP^n°j: sin^33j (sin 45) 



sin 67^ 



1919. 



B/22iv<=s^"'78°| sin256°J (sin 46)* 

 ^ ""' sin67°i 



p/ggi u_sin' 56°| sin 67°^ (sin 46) 

 '^ ^^ 2 sin278°| sin222°i 



C(22M<=^^"*^^°li^" 67°| (si n 45) 

 ^^ 2 8in2il°isin222°i 



