186 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1912. 



VIII. 



The majority of the equations whereb}'' we inter})ret the physical 

 phenomena have certain properties of continuity. The soUitions 

 vary in a continuous manner, at least during a certain interval 

 greater or less in length, when the given quantities vary in a contin- 

 uous manner. Besides, this property is not absolutely general, and 

 it might happen that the theories of the quanta of emission or absorp- 

 tion may lead to attaching more importance than has been done here- 

 tofore to exceptional cases; but to-day I do not wish to begin this 

 discussion. I rest content with the general property, verified in a 

 very great number of cases. 



When we seek to interpret this property in the theory of the poten- 

 tial and of the monogenic functions, we should expect, if for simpH- 

 fication we confine ourselves to the real functions of a single variable, 

 to find a sort of continuous passage between such of these functions 

 as are analytical in the Weierstrassian sense and those wdiich are en- 

 tirely discontinuous. But it is tliis which does not occur unless we 

 consider nonanalytical monogenic functions. From tlie moment 

 when a function ceases to be analytical, it no longer possesses any of 

 the essential properties of analytical functions; the discontinuity is 

 sudden. The new monogenic functions permit one to define func- 

 tions of real variables which might be called quasianalytical and 

 wliich constitute in some way a zone of transition between the class- 

 ical analytical functions and the functions wliich are not determined 

 by the laiowledge of their derivatives in a point. Tliis transition 

 zone deserves to be stucUed; it is often the study of hybrid forms 

 which best teaches in reference to certain properties of clearly deter- 

 mined species. 



We see that the points of contact between molecular physics and 

 mathematics are numerous. I have been able only to rapidly point 

 out the principal ones among them. I am not competent to ask 

 whether the physicists will be able to derive an immediate profit 

 from these analogies; but I am convinced that the mathematicians 

 can only gain by going into them thorougldy. It is always by a con- 

 tact with nature that mathematical analysis is revived. It is 'only 

 because of this permanent contact that it has been able to escape 

 the danger of becoming a pure symboUsm, revolving in a circle about 

 itself; it is oAving to molecular physics that the speculations on dis- 

 continuity are to take their complete signification and be developed 

 in a way really fruitful. And, for lack of exact apphcations impossi- 

 ble to foresee, it is sufficiently probable that the mental habits cre- 

 ated by these studies wiU not be without advantage to those who 

 shall desire to undertake the task, wliich wiU soon be imposed, of cre- 

 ating an analysis adapted to theoretical researches in the physics of 

 discontinuity. ^ 



