52 REPORTS ON THE STATE OF SCIENCE. — 1919. 



For trisection in general, a;=/csn^|K is a root of the Jacobian 

 quartic 



a;=N/(l + c)+v/(l + a)c)+ v/(l + oj2c), a)"=l, c-'=^r -^j . 



K iX \-^ K J iX 



_{l-xY{3 +x), or {l+xf{S-x) 

 4^ ' 



„ , - 2,^ b-1 ^ 2„ 6-1 

 So also, for cn^K = c~: -i ' or dn3E. = tr ' 



d 0+ i tJ ■^ 



Thus, for the lemniscate function, with c=^, 



1,2 /3 /3+y3 ' /3 ^ „n/3+1 



'^^V=-V 2+\/ -2" = -V 2 + ^^"^^' 



Ji_6&2_3=0, 6=</3'^^^\ 



1 V cos 30 _sin45(sin60)i 



^^3-'^= V 1+ cos 30- sin75 ' 

 1^ V3 + 1+^^2V3 ,2^ _V3 + l-y2V3_ V/l^lineON 



2 "V U+\/sin60/ 



sIgL= 2 , cl3L = 



-.1^ V3- 1 + N/2V3 2 ,,ov/3 + l 1 

 ^k^ = 2V2 ' ^h^ = ^ ^ 2^2^-2" 



Quarter Section : ?'=225. 



4. Taking the formulas in Phil. Trans. 1904, p. 278, for /x=8, and 

 changing to the angle a=^(45), cot a=s/K, tan^a=a, the expressions 

 may be deduced 



^ ^^ Sin(j7r + a) 



-Qf22i.Y=^^^3l±M cos'ia(sin 2a) ' 

 ^ ^^ sin(J7r + a) ' 



D(22i)4=cos2ia sin (l7r + a)(sin2a)'^ 

 ^ ^' 2 Sin2(j7r + la) COS^a 



and then 



P,ooi\4_sin24a sin + a) (sin^a) -' 

 ^ ^^ 2 sin^d-Tr-ia) COS^a ' 



k' tn^iK=-^igi)J=tan (i^-ia) tan ^a, 



dn2JK_C (22i)^_ tan( i7r- ^a) 

 k' ■-D(22i)2 tanW ' 



