472 



SCIENCE 



[N. S. Vol. XLV. No. 1168 



and the corresponding expansions will no 

 longer serve to yield us tlie most satisfying 

 form for the expression of our results. 



This situation has been contemplated 

 with uneasiness in certain quarters. To 

 some natural scientists it has seemed like 

 the loss of our moorings. To some mathe- 

 rnaticians it has appeared in the light of 

 greatly lessening the importance of many 

 investigations, difficult and prolonged. It 

 is said that Poincare contemplated the out- 

 look with keen regret. But we had as well 

 make up our minds to the situation. It 

 seems almost certain that electricity is 

 done up in pellets, to which we have given 

 the name of electrons. That heat comes 

 in quanta also seems probable. In fact, it 

 is not unlikely that we are on the verge of 

 interpreting everything in nature as essen- 

 tially discontinuous; and it would perhaps 

 be no surprise to us now to find that even 

 energy itself is not unlimitedly divisible, 

 but exists, so to speak, in granules which 

 can not be separated into component parts. 



A few years ago such a paragraph as the 

 foregoing would have been thought a piece 

 of nonsense and to be not entitled to con- 

 sideration; now the author is more likely 

 to be charged with repeating something 

 which already has been heard to the point 

 of weariness. 



In view of so sweeping and fundamental 

 changes in our outlook, what is going to 

 happen to the existent body of applied 

 mathematics? Simply this, if these new 

 ideas gain currency: that which before 

 had been considered a very close approxi- 

 mation to facts will now be treated as giv- 

 ing only a coarse first approximation; and 

 we shall set about the task of finding means 

 of studying phenomena more exactly in 

 consonance with the new underlying ideas. 



You will probably be disposed to ask in 

 what direction we shall turn now to find 

 the requisite mathematical tools and when 



we can expect to have them ready for use. 

 It may be answered that the mathematician 

 was beforehand with a partially developed 

 tool which will probably serve the pur- 

 pose. When these new ideas in physics 

 were just coming to the front a few young 

 mathematicians independently of each 

 other and apparently without knowledge 

 of these movements in physics were en- 

 gaged in the study of certain mathematical 

 problems having to do with a thing which 

 wiU probably turn out to be a suitable tool 

 for the investigation of discrete phenomena. 

 At any rate, the equations which they were 

 studying are not intimately bound up with 

 considerations of continuity as are differ- 

 ential equations, but yet they possess a 

 number of properties very similar to or in 

 common with those of differential equa- 

 tions. The equations which are thus 

 brought to a new position of importance 

 are the so-called difference equations. 



Simple difference equations first ap- 

 peared in the literature rather early in the 

 history of mathematics and certain ele- 

 mentary aspects of their theory were con- 

 sidered several generations ago. But in re- 

 cent years an essentially new type of prob- 

 lem in connection with them has come to 

 notice; and in a short time and through 

 iseveral independent investigators the theory 

 has suddenly blossomed forth into unex- 

 pected and magnificent flower. 



This development had its origin almost 

 simultaneously in three countries and in 

 the hands of three independent investi- 

 gators: Norlund in Sweden, Galbrun in 

 France, and myself in America. My own 

 first contribution was followed closely 

 (also in this country) by Birkhoff's first 

 fundamental paper in the new field. By 

 this time numerous other persons have 

 made contributions to the development of 

 the subject both from the function-theoretic 

 point of view of the papers just mentioned 



