ON THE CALCULATION OF MATHEMATICAL TABLES. 61 



Lemniscate Seven Section. 



6. With the Weierstrass functions of the First Stage, as defined in 

 Phil. Trans. 1904, p. 250, and in the 2?z + l section of a period, 



(1) ,.,„,=0, A=^^., X.=r^; /=^, r=.90/. 



_1 1 



(2) A(2r) =i/{e,-e^.e,-e^)x »A ^"^'' 



_i jt^ 

 (8) A(2^r)=V (61-^2 • e,-e^)x 'A-='"+V' 

 but still requiring the condition of the Second Stage, that the factors 

 of S should be known. 



But with the Lemniscate Function, where 



(4) gr3=0, A=sr23=(«,-e3)«=64(e,-e2)«=64(e2-C3)« 



(5) Ki2r)=i\^fi\~'^\ 



Li2pr)=.(^^^Y\~K~'--\,. 



Turning to the case of 2w + l=7, Phil. Trans., §9, p. 230, and Proc. 

 L.M.S., 1893-4, p. 223, where Klein's modular equation is 



J:J-l:l = (T^ + 13r + 49)(TH5r + l)3 : (T^ + UfHeSr-' + TOT-T)^: 1728t, 

 ^^l-8z + 5z-^ + z^ ^13^(1+^) . ( 2-.Z) . (l-2g) 

 z{l—z) ' "^ 2 2z . 1—z ' 



\ 2 J 4 z^{l—z)- 4 



r + ^^=l-z-±—'-^=:l^3cot Se; 

 2 2 1—x z 2 



and then, according to Mr. Alfred Lodge, the three roots of this cubic 

 equation for z, when t is given and the auxiliary angle 0, are given by 



^ sin ^ J__sin(60+^) 2-l_sina20 + i9) 



sin(60 + ^)' 1-2 sin(120 + (9)' z sin(18O + 0)' 



The lemniscate condition, J=l, requires 



t< + 14t3 + 63t2 + 70t-7=(t2 + 7t + 21)2-7(2t + 8P=0, 



r= -'^y^'^ + ^^1^^7=0-09219 27 .... 



2t + 13=13-1843854, cot 36l=al 0-40437bO=cot 21° 30'. 606 



sin7°10'.202 , f-0960644 i i iqicoqc aiqk.oqq 

 ^=sInW°T0'.202=^^ 1-9645709=^^ M316935=0-1354233 



1 _ sin 67° 10' . 202^^^ 1:9645709^^1 00681965=M566355 



\—z sin 127° 10'. 202 1-9013744 



.-1^ sin 127° 10'. 202 1-9013744^^1 0-8051100=6-384251 



t -sin 187° 10'. 202 1-0960644 



