( 854 ) 
the boundary of the two domains, to enter at G the other domain g, 
and leave it no.more. 
Let now y' be=—=4y and A’, (coinciding with H), A’,, A’,,.... 
a series of points on 7 in such a way, that each are A’,A’n41 
is a y’-are. In the same way as we have constructed the curve 4, 
we now construct a single closed curve 4’ consisting of an are GH’ 
of 7, and an arc of principal circle G’H’, which is smaller than y’. 
This curve 4’ lies entirely inside g,; for after the preceding the 
only way in which it might still leave it, is that the two ares of 
principal circles GH and G’H’ should meet in two points, which 
is impossible, both ares being <2. 
So the curve &’ divides g, into 1. an annular domain (which 
only in the special case that M and G’ coincide becomes singly 
connected), in which lies the are HG’ of 7, and 2. a singly connected 
domain g',, within which lies the pursuing branch of 7 past H’. 
Repeating this process indefinitely and taking every time y+) = 3 y, 
we construct a type of order w of single closed curves, of which 
each following one lies within the preceding ones, and it is easily 
proved, that as soon as y™ has fallen under a certain maximum }), 
the following curves 4 have all a length smaller than (p + 3)yV 2. 
Hence the lengths of a// curves 4% lie below a same finite limit. 
We can now regard the place on the sphere of a variable point 
of © as a function of the length of arc s between G“ and that 
point. The different ks are then represented by a system of 
uniformly continuous functions. Thus according to ARzELA’) a 
fundamental series k 7 
Km), k(t2), KD), 
can be indicated, converging uniformly to a continuous limit function k. 
The differential quotients of the functions determining the curves 
k©) are in any point indicated by the vector direction in that point; 
they are uniformly approximated by the functions of s determining 
the difference quotients with respect to s, and der are themselves 
uniformly continuous functions of s 
So they converge uniformly he ae limit function represent- 
ing the differential quotient, i.e. the tangent direction of £@. 
1) Such a maximum is the quantity 5 mentioned in the preceding note. For if 
we then take Gi) as Ag, and if we construct the corresponding curve k, then 
the length of the arc between G(™ and the point H belonging to that curve k is 
smaller than (p +2) yWV2. We know here however for sure, that, if before this 
pomt H no point on 7 has been reached possessing a distance smaller than 5 from 
Gm), such a point will not appear farther on either. 
2) „Funzioni di linee”, Rendiconti Lincei (4) 5, 1 (1889), p. 342; “Sulle funziont 
di linee’’, Memorie della Accademia di Bologna (5) 5 (1895), p. 225. 
