ON THE CALCULATION OF MATHEMATICAL TABLES. 51 



(sin 45)5=al 1-92474 25011 

 sin I5=al 1-41299 62306 (sin 15)' -al 1-11949 43459 



1-04423 68470 



(sin 60)1= i-98438 26579 



E(30) = ali-05985 41891 



= 0-13253 28561 



(sin 45)? = al f-92474 25011 

 (sin 60)! = al 1-98438 26579 

 (sin 75)*= al 1-99247 18890 



a=.(sin 45)! (sin 60)! (sin 75)?=al 1-90159 70480 



= 0-79725 46262 

 (sin 45)?=al 1-77422 75033 

 (sin 75)i=al 1-97741 56671 



(sin 45)? (sin 75)?=al 1-75164 31704 

 (sin 60)'; = al 1-92191 32897 



/5 = (sin 45i) (sin 76)*=:al f-82972 98807 

 (sin 60)1= 0-67566 26010 

 F{30) = a-/3 



= 0-79725 46262 

 — 0-67566 26010 

 = 0-12159 20252 



This checks all the trisection values in the lemniscate table ; but 

 some other corresponding values of the elliptic function may be 

 cited here. 



Among all the trisection values for the different modular angles, the 

 simplest appear to arise for 6=75° and there 



*¥' = V3=V(si^60°)' ^"3^^= ,/2=«i°45S 



D(30) ®3^^ 1., ^, , 1 1 2^2 



D(90)= ®K -2^^' ^(^^)= v'K' = ^(sTnl5)'^(^^) = sT(sin"l5°)' 



A(30)=^|---\ C(30)=^^ + \ B(30)=^2V3y(sin 15), 



E(30)=,^3, F(30)=^2^3 ■ 



The Table for a=75°, K=KV3 is given in Report 1912, p. 52 ; and 

 it might have been derived by the cubic transformation of p. 89, Report 

 1913, applied to the Table for 6=15°, K'=Kv/3, p- 48, Report 1912, for 

 which a q series expansion is rapidly convergent. 



These division values are useful in settling the number of terms to be 

 employed in the series. 



