Mat 18, 1917] 



SCIENCE 



473 



and also from another direction with a 

 consequent consideration of a different type 

 of problem. 



So far the recent workers in this field 

 have given their entire attention to purely 

 mathematical developments and have not 

 considered so much as the possibility of a 

 use of the results in the domain of applied 

 mathematics. In particular, my own in- 

 terests have been in theoretical considera- 

 tions. But I look forward to important 

 applications of this newly developed theory 

 both because it seems to have in it some at 

 least of those elements which are necessary 

 to accord with phenomena which are dis- 

 continuous in their nature and more par- 

 ticularly because there is here an expan- 

 sion theory also consonant with the dis- 

 continuities of nature and related to differ- 

 ence equations in a manner somewhat sim- 

 ilar to that in which certain other expan- 

 sion problems attach themselves to differ- 

 ential equations. But the analogy must 

 not be pressed too far, since there are also 

 essential differences. 



Concerning these new expansion prob- 

 lems I wish to say one further word. It 

 is very recently indeed that they have come 

 to notice; and a knowledge of them is not 

 yet generally current. In fact, the general 

 definition of the series involved was first 

 made in a paper of my own published less 

 than a year ago; and I am still engaged 

 in working out their more detailed theory. 

 Doubtless other workers in the near future 

 will take up different phases of the same 

 problem. 



So far no exposition of the modern 

 theory of difference equations exists in the 

 literature ; the results are to be found only 

 in the original memoirs. In a few in- 

 stances this theory has furnished the basis 

 or an integral part of a course of academic 

 lectures. As • such it has appeared, as I 

 understand, in one of the courses at Har- 



vard. I have myself delivered lectures on 

 it in Indiana University and the Univer- 

 sity of Chicago ; and it is my purpose next 

 year to expound this new doctrine in my 

 lectures here. It is highly desirable that 

 this matter shall be developed rapidly and 

 be prosecuted from various points of view. 

 It is in this way only that we shall be able 

 to learn what its import really will be for 

 the progress of science. 



Before concluding my remarks I wish to 

 speak briefly of a different sort of concep- 

 tion or expectation which has arisen in 

 some quarters and having to do with a more 

 fundamental and far-reaching use of 

 mathematics than any yet made. It is 

 connected with the fact that every branch 

 of physics gives rise to an application of 

 mathematics and the consequent feeling 

 that there must be a deep underlying rea- 

 son for this and a consequent close relation 

 of phenomena which probably makes them 

 capable of an explanation from a single 

 point of view consistently maintained. 



If there is a "hypothetical substructure 

 of the universe, uniform uxider all the di- 

 verse phenomena," it would appear that 

 there must be some means of ascertaining 

 what it is and of giving to it a mathemat- 

 ical expression and body. At any rate the 

 expectation of such a thing has arisen ; let 

 us hope that the event will show that the 

 anticipation is well grounded in the nature 

 of things. 



I understand that the earliest contribu- 

 tions to just such a development are al- 

 ready in existence; that the now current 

 theoretical accounts of radiation, diffusion, 

 capillary action and molecular behavior in 

 general have just such characteristics as 

 one would expect to find in the early stages 

 of a mathematical theory of the substnic- 

 ture of the universe. 



Let me guard against a misapprehension 

 concerning the foregoing remarks. I have 



