184 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1912. 



and continuous, this potential satisfies the equation of Poisson, which 

 reduces itself to the equation of Laplace wherever the density is zero; 

 that is to say, in all the points of the planes which we have just 

 defined. It v/as not profitless to insist upon this point, for these 

 planes may traverse regions of space in which the given material 

 points are everywhere dense, as are, for example, all the points whose 

 coordinates are rational numbers. We might have entertained a fear 

 lest there should not be free space between the points so much pressed 

 together in any manner whatever; we have just seen that this fear 

 is not justified. The theorem of the theory of ensembles which is 

 necessary and sufficient for demonstratmg this result in a rigorous 

 manner is the following : If on the segment of a straigM line we have an 

 infinite number of imrtial segments (in space, the projections of spheres 

 of action) whose total length is less than the length of the segment, there 

 exists on this segment an infinite numher of 2>oints which do not pertain 

 to any of the jpaHial segments. This enunciation is almost evident, 

 and, besides, it is easy to rigorously demonstrate. 



In tlic case of the plane we shall replace the spheres by circles and 

 the plane perpendicular at a pomt of the segment by a perpendicular 

 straiglit Imc. We easily prove that, even in the region where the 

 singular pomts are everywhere dense, there are pomts in which an 

 infinite number of such Imes intersect at which the density is zero. 

 In these points logarithmic potential function satisfies Laplace's 

 equation in two variables. If we study in a similar v\^ay the function 

 of a complex variable with poles dense in one region, we define m 

 this region an infinite number of straight lines of continuity inter- 

 sectmg hi all directions, the function admitting derivatives which 

 are continuous on these lines, and the derivative having the same 

 value in all the directions in each of the points of intersection. For 

 expressmg this fact we shall employ the expression created by 

 Cauchy for dcsignatmg functions which admit a derivative inde- 

 pendent of the argument of the increment of the variable. These 

 functions will be called monogenic; but they are not analytical, if we 

 reserve for the word "analytical" the very precise meaning which 

 it has possessed smce the labors of Weierstrass. 



Without lingering on the physical analogies suggested by the 

 existence of planes which do not intersect the spheres of action of 

 the attracting masses, I wish to insist a little upon the nature of the 

 mathematical problems set by the existence of the monogenic but 

 nonanalytical functions. 



We know that the essential property of analytical functions is 

 that of bemg determinate m their whole domain of existence when 

 their values are given m one portion, however little it may be, of 

 this domain. Is this property a consequence of analyticity — that is 



