MR. E. T. WHITTAKER ON THE CONNEXION OF 
2 
been studied by Schottky,'" Weber,! and Burnside, ;|; and are capable of uniformising 
any algebraic form. As, however, the fundamental polygon is multiply-connected, 
the Abelian Integrals of the first kind, and the factorial functions associated with the 
ajo’ebraic form, are not uniform functions of the new variable. 
With regard to the analytical connexion between the uniformising variable t and 
the variables u, z, of the algebraic form, Poincare proved that if 2 is an automorphic 
function of t, then {t, z] is another automorphic function of the same group, where 
{t, z} is the Schwarzian derivative, t therefore satisfies a difierential equation of 
the form 
{t,z} = 4>{u, z), 
where (Jj (u, z) is some rational function of u and 2. Schottky and Weber have 
determined (p [u, z), save for a number of undetermined constants, for the groups 
found by them, and Klein § has obtained more general results, applying to any 
algebraic equation, but with a certain number of undetermined constants left in <^. 
The problem has been formulated by Klein as one of conformal representation. 
The algebraic form which is given by 
f{i(, 2) = 0 
can be represented on a lliemann surface of class ^3, so that, corresponding to every 
pair of values {u, z) of the form, there is a place on the surface. By drawing 2p cuts 
we can make this surface simply-connected. Now let 2 be regarded as a function 
of a new variable t, having the following properties :— 
1°. The dissected Biemann surface is to be conformally represented on a plane 
area in the ^-plane, bounded by Ajj curvilinear sides (namely, the conformal repre¬ 
sentations of the cuts, each cut giving two sides). 
2°. Of the two sides of the i-area which correspond to any cut, one is to be 
derivable from the other by a projective substitution 
[f 
\ ’ + dj ' 
3°. The group formed by the combination and repetition of these 2p substitutions 
is to be discontinuous. 
A¥hen a variable t has been found satisfying these conditions, u and 2 will be uniform 
automorphic functions of t ; and we know by the existence-theorem of Poincare and 
Klein that such a variable does exist, although the existence-theorem does not 
connect it analytically with 2 and u. The primaiy result of the present paper is, 
that the uniformisation of any algebraic form can be effected by automoiq^hic func- 
• * * * § ‘Crelle,’voL lOI, 1897, p. 227. 
t ‘ Goltinger Nachrichten,’ 1886, p. 359. 
t ‘ Proc. Loud. MatE. Soc.,’ vol. 23, 1891, p. 49. 
§ ‘ Jaliresbei’icEt der Deutsclien Llatliematiker-Yei'eiDiguug,’ 1894-5, p. 91. 
