68 REPORTS ON THE STATE OF SCIENCE. — 1919. 



Lemniscate Five Section, again. 



8. A return can be made to Five Section in this method of the First 

 Stage, as in Phil. Trans. 1904, p. 229 ; there with 



75=0, y=x, X=y'^=x^, 



and the modular equation 



J : J-1 : 1=(7^-10t -1-5)3 : (T2-22r + 125) (t2-4t-1)2 : -1728r, 



J=l, t2-4t-1=0, t=2-v/5, ^-a;=ll-T=9+ V5=8-|-4 sin 54 



X 



^+a;=12sin72° 



X 



x=Q sin 72-4-2 sin 54° 



= I6sin29 sin 39° sin 51°=al2-9459 .... 

 A=-«6^-a;«(v/5-2)=a;«(2 sin 18)^ 



A(2r)=a;"^X"'^Yl ^WV^jjVo /gin igVi 



«i'»=al i-89459 . . . 



(sin 18)^=al 1-87250 . . . 



A(2r)=al 1-76709 . . . =0-5849 . . . =A(36) 



AL*i:)=a;~i=al 0-2108 . . . 

 A(2r) 



A(2r) =al 1-7671 



A(4r) =al 1-9779=0-95 =A(72) 



ij."« J-a;{2sinl8°)* ^(2 sin 18)* 



cl>3 L= ^ — ^ ^ = ^ 



l^^l+xf + ^x] ,^xQ+. + 6) sin72+l 



^(2 si 



sml8°]^ _sinH5° (sin 18°)* 



"2 sin 51° sin 69° sin 51° sin 69° 

 . . =al i-7164 . . . =cos 58°.6=cos^(54) 



