10 
MR. E. T. WHITTAKER ON THE CONNEXION OF 
that the method about to be given for the division of the plane into regions breaks 
down ; but if n > 2 , the relation can he satisfied, in an infinite number of ways, by 
substitutions of the required kind. A worked-out example is given below. 
[Added June 2, 1898 .—The possibility of the construction given below depends on 
the satisfying of certain inequalities among the constants of the substitutions ; as in 
general, when the construction described is carried out, the sides of the polygon may 
cross each other.] 
Now let Di, D2,. . . D„+2 be those double points, of the substitutions Sj, Sj, S3, .. . S,i+2 
respectively, which are above the real axis. 
Let Cl be the point derived from 1)^+2 by applying the substitution Si ; or, as we 
can write it, let 
Cl = Si (D„,2). 
Similarly, let 
C2 = S2 (Cl), C3 = S3 (C2), . . . , C,.,1 = S,.,1 (CJ. 
Then 
C„+1 = S,,+iS„. . . S2S1 (D„+2) 
= S,,+2 (D^+o), since S1S2 . . . S,,+2 = L 
— D',1 + 2* 
Now, by the last theorem, any point, and the point which is derived from it 
by a self-inverse substitution, lie on a circle through the double points of the 
substitution. 
Therefore D„4.2DiCi lie on a circle orthogonal to the i-eal axis. 
Similarly C1D2C2, C2D3C3, . . ., C,iD„+iC,j+i, all lie on circles orthogonal to the 
real axis. 
Therefore a curvilinear ^jolygon can he formed, whose {n + 1 ) sides are arcs of 
circles orthogonal to the real axis and pass through the points Dj, D2, D3, . . . D„+i, 
respectively, and ivhose corners are the points I)„+2) Ci, C2, . . . C,j. 
Now suppose we transform the polygon by the substitution S„ where r = 1, 2, 
. . . (n-|- 1). We obtain another polygon, likewise formed of arcs of circles 
orthogonal to the real axis, and having contact with the original polygon along the 
side C,._iD,,C,.. The side of this new polygon which is the conformal representation 
of C^_iD^C^ passes through the double points of the self-inverse substitution S^.S^S,.; 
and on applying this substitution to the new polygon, we obtain a third polygon, 
having contact with the second along the side which is the conformal representation 
of Cp_iD^C^. In this way we can, as every new polygon is formed, surround it with 
other polygons, each having one side in common with it. 
Now consider what happens at any angular point of the polygon, say D,j+2) when 
we derive polygons in this way. If we derive a fresh polygon by applying the 
substitution Si, the derived polygon adjoins the original one along the side D„+2Ci. 
It now we derive a fresh polygon from the original one by applying the substi- 
