308 REPORTS ON THE STATE OF SCIENCE, ETC. 
Thus 
taken over the wnole area of the circuit. 
Similarly 
fe: 5 (do 
—! dy= — ) dad 
is 5 ay le (G,) aurdy 
Thus 
r (8 (3m 5 /80\ ) 
2— 2 a9) kay (ae lad 
@).-@ =| {ie (32) - 5 (52) pores 
=O 
if inside the circuit stp exists and is defined in an unambiguous manner. 
Now it is clear that if 
| : Be . ds +{ ae ds =0, 
APB ds BQA bs 
then, reversing the sign of the second integration 
| PP ds = \ bb 4 
APL ds AQB oe 
($y — 4) along AvB= ($y — PA) along AQB, 
that is we obtain the same value of > at B if we build it up by integration along 
any two reconcilable paths, starting from a definite selected value at a point A. 
The value of @ thus obtained at any point which can be reached from A by a 
continuous path is thus defined in an unambiguous manner, provided the connectivity 
of the path, i.e., the manner in which it winds amongst the holes in the material, 
is specified. 
Now, if A is a point of a reducible circuit, all paths AR from A to a point inside 
or on the circuit, are clearly reconcilable. We thus get a function ¢ unambiguously 
bo 
or 
defined in the same manner that * and were unambiguously defined. 
ac 
y 
It follows that we can in this way Soe eae the function ¢ from its first 
By and aa exist and can be unambiguously 
differential coefficients, provided ae by 5237 
defined in the region considered. 
If we continue our function ¢, obtained in this way, round an irreducible circuit 
(enclosing say a hole in the plate), then on reaching again the same point A we 
get, in general, a value ¢, different from ¢,. We get, however, the same value 9, 
for any “other irreducible circuit reconcilable with the first one, so that #, is a definite 
function ot the co-ordinates z, y of A determined by the connectivity of the 
irreducible circuit. 
We shall call $.—9, the cyclic function of » for this particular connectivity and 
denote it by Cy(); it will usually be convenient to omit the indication of the 
circuit typical of the connectivity: this may be specified by a suffix denoting the 
circuit in question, where required. 
It will follow directly from the definition that 
80) _ 8 ¢ 
Cy (sr) = 5p 7 
so that Cy and a =, (o : ) are commutative symbols of operation. 
y 
5. In what awe the stresses and strains will be restricted to be essentially 
one-valued, that is, no restriction has to be placed on the functions which represent 
them in order to make them one-valued over a given region and this quite 
irrespectively of multiple connection of the space considered. 
