ON THE CALCULATION OF MATHEMATICAL TABLES, 65 



Lemniscate Nine Section; r=10. 



7. Here, in L.M.S. 1893, p. 233, with ^+3=a;, the modular equation 

 of the 9th order becomes 



J : J-1 : l=xV-24)-^ : (x6-36x-^ + 2]6)'^ : 1728(a;=*-27), 



so that the lemniscate condition, J=:l, requires 



a;6_36^3 + 216=0, a:»=18 + 6v/3, x=n/3^(2s/3 + 2) 



a;=al 0-4844002 =3-050705 .... 



Then from Phil. Trans. 1904, p. 231, § 10, 



y,=o, x=f-{i-p){i-p+p% y=p\l-p), A=.^=l-^=2P(l-p), 



A=jy-'(l-_p)i-^(a;^-27), with x3-27=6s/3-9=l-3923048 



^A=p'-'{l-py^a\ 1-2406443 

 8 



'•9367280 



(^^^y'=p{l-p) all'- 



The cubic for the parameter p is here 



p{l—p) 1—p p 



^2 p{l-p) 2 



^^^oot36=^+P^-^-^4^^ =-«,-! = -4-550705 . . 



from which cot (180°-3e)=al 0-2434268 



.-. 3^-180°-2y° 43' 22"-06 • 



6=50° 5' 32"-65 



P= ■ !!.^^ .. =al 1-9121104=0-8167900 



^ sin(60 + ^) 



-L= s""'(60 + ^) =al 0-7370511 = 5-4582210 



1-^ sm(12O + 0) 



— ^i = -^^(M+^ =al 1-3508385=0-2248048 



p sin(180 + 6) 



ri(a;3-27)y^=al 1-9367204 



