42 Proceedings of the Ro>jal Irish Academy. 



A type of path from 6 to - is represented in figure 2, wherein is the 

 origin of Q, and A, A' are points in the axis equidistant from 0. The path 

 from ^ to ^' is in the axis, except for semi-circtdar detours round branch- 

 points between - c and c, and a circular detour round c, which may be a 



branch-point. 



i6 



^' P ^ A[ .^^ 



-e' 



FlGCRE 2. 



The function ^(0) may have one or more groups of branch-points, such 

 that within each group the powers of the branchings are additive.* Attention 

 being directed to the branchings of such a group, it is known that each semi- 

 circle of detour round a point where there is branching of power a introduces 

 a factor erp (- iVa) into the corresponding part of the function. Now A may 

 be taken anywhere from to c, and if it be possible by moving A to introduce 

 or remove a semi-circular detour in the part of the path from ^ to c without 

 making simultaneously a corresponding change in the semi-circular detours in 

 the part of the path from to A', then it is impossible for the effect of the 

 traversing of the path from A to A' to be independent of the position of A, 

 as it must be if (7iA^ = (s(,A). Hence it is necessary that, within each 

 additive group, the distribution of branch-points along the range from -doe 

 be symmetrical with respect to 0. 



This symmetry once recognized, it is seen that to each branching of power a 

 in the range from to c there correspond two semi-circular detours in the 

 path, either two at the same jwint, if between A and c, or two at points 

 symmetrically situated with respect to <?, if between A' and A. The only 

 e.xception is the branch-point (if any) of power o,> at 0, for which there is 

 only one semi-circle. As the function is to have the same value for - 

 as for 0, the cumulative effect of all the semi-circular dfetours corresponding 

 to the branchings of an additive group must be the restoration of the original 



• This may be explained by an example. In the case of the function 



9"--f 1 + (*«- «*)i-r (e-»)^ (9 - «}i (8 - djk, 



the powers of the branchings at e = - o may be regarded as additive, since they affect 

 the same term of the function : and the branchings at e = e. i = e. » = d form a group 

 whose powers are likewise additive- 



