528 TRANSACTIONS OF SECTION A. 



where 



*, > <£+* f L l < £+£, 2* - 2* < £+* 2, -2 -1 > 2+2\ 



N N N N 



and so on, just as before, and the number of terms in the value for - — zJJP { s evev 



N 

 when e= + 1, and otherwise odd. Then putting 



h, = {(2*- 2*»+ • ■ • +2 /i2 '-i)N-2'^J2 -M 2'-i; 



/'2 S + i=|-(2"'-2"=+ . . . -2 U 2»)N + 2'^}2~' X ^, 



we find as before that the cycle beginning with the number p, and proceeding as 

 before according to the rule j = 21, orj = N — 2%, is given by the series 



7) 22? . - . 2' — **» — ^iy 



N-2'-"'j>, 2(N-2'-^j»), . . ., 2".-"a- 1 (N-2'-".) 



K 3 , 2K 3 , . . ., 2"=-".-iK 3 , 



K 4 , 2K 4 , . . ., 2^-^-iK,, 



• ■ ., 



and consists of t numbers. And it is not difficult to show that t = r or is a divisor 

 of r. 



For an example of some length it may be verified that when n is 412 we have 

 nineteen cycles of each 20 numbers, one cycle of 10 numbers, three cycles of each 

 5 numbers, one cycle of 4 numbers, one cycle of 2 numbers, and one number 

 unaltered throughout a column. Thus there are 20 permutations in the group 

 formed with 412 letters, and we have the partition 



412 = 20 + 20 + ... (nineteen times) +10 + 5 + 5 + 5 + 4 + 2+1. 



3. On the Trisection of the Elliptic Functions. 

 By Dr. H. F. Baker, F.R.S. 



A quartic equation for which the two conditions are satisfied, (1) that the 

 invariant of the second degree vanishes, (2) that the sum of the roots is zero, may 

 be supposed to have this form : — 



1 , 1 



q„x' — <7,.7• — 



2 y2 J3 48 s 



f(x) = x* - n g^x 2 - g 3 v - ; o <jI = 0. 



The root of this equation may be supposed to be 



.r, = A(l-/0 2 , .r., = A(l-/*e) 2 , ,r 3 = A(l-,ue 2 ) 2 , x t = -3A, 

 where e is an imaginary cube root of unity, and /u, A are such that 



p»Qi' + 8 )'_ 27g|-g| A= 12* 1-n* 



4\L-^y g\ ' 9i S + 20m 3 -m 6 ' 



Hence these roots are connected by the two equations 



\/x, + e \/.r., + t- \ / x 3 = 0, S.r, + e 2 Vx, — e Vx., = 0, 



besides twa others linearly deducible from these. 



If the original quar.ic equation be transformed by putting so' = (px + q)(rx + s), 

 the quadric invariant of the new equation in a; 1 will also vanish ; if the sum of the 

 four values of as 1 is also zero, it is easily found that x y is an arbitrary multiple of 



A — a; 



in which X is arbitrary, and/'(X) = 4X S — g 2 \ — g 3 . Thus the square roots of the four 

 expressions obtained from this by replacing in turn x by a?„ x 2 , x 3 , x t , are likewise 

 connected by two linear equations. A particular case is obtained by taking X equal 

 to one of the roots of /"(A.) = 0, say e ; then we obtain the equations 



1 £2 =o, - L + _Jl_- -.^-,=o. 



Vx x — e </a\,— e </x 3 — e \/x t — e •/x l — e Vx. 



