30 
MR. E. T. WHITTAKER ON THE CONNEXION OE 
In general, tlierefore, we have 
T = [ttiRi (2, u) + ttoR. {z, w) + . . . + a3^_3R3^_3 {z, u}] {dzjdtf, 
where Ri (2, u), R2 (2, u), . . . R3 (2, u) are functions which can be found, and 
«!, . . . GLzp-z, are arbitrary constants. 
We can now fiyid the form of the differenticd equation ivhich gives all the quasi- 
uniformising variables. Take any quasi-uniformising variable t of the algebraic 
equation 
f {u, 2) = 0. 
For it, we have 
^ 1^5 z } — ^ { Z } ^)} 
where (/> is some rational function of 2 and u. 
If t is the most general quasi-uniformising variable, we have seen that t is given as 
the quotient of two solutions of the differential equation 
dh^ldd + Tv = 0, 
where 
T = [aiRi {z, u) + a2R2 (2, m) + . . . + (2, u)~\ {dzjdtf. 
Hence 
ifbr} =T. 
But 
{t,z} = (t, 2} + {drldzj {t, t]. 
Therefore 
^ {t,z} = (f){z,u) T {dTjdzf, 
or 
^{t,z} = (f> (2, u) -b aiRi (2, ic) -f a^R. (2, w) -f . . . + Oa^-sRsp-s {z, u). 
Thus, the solution of the problem of finding all the variables t, which ivill con¬ 
formally represent the Riemann surface of a given algebraic form on a curvilinear 
polygon, ivhose sides are derived from each other in pairs by projective substitutions, 
is given by a differential equation containing {^p — 3 ) arbitrary parameters linearly, 
and the problem of finding the unifio rmising varicdde is equivcdent to tha t ofi deter¬ 
mining these parameters in order that that group generated by these substitutions may 
be discontinuous. 
Now let us return to the differential equation of § 5 , which we can write 
1 
2 
{t, iv] 
8 (n 1 ) u diih 
1 
-—■R hz-^ + hz-^ + • 
. , -f- ^ 
If we take any set of values ^2 . . . kn_i for the undetermined constants, 
this differential equation will give a variable r in terms of 2, which will not in 
