172 ANNUAL EEPORT SMITHSONIAN INSTITUTION, 1912. 



finite number of periods superior to a given interval, but tliis number 

 increases indefinitely when the interval tends to zero. 



Tbe reasonbig wbicli has just been outlined is of the type of those 

 to which the substitution of continuity for discontinuity leads. In 

 reality, the considerations based on the existence of molecules play hi 

 them only an auxiliary role; they put one on the track of the solution, 

 but tliis solution, once obtained, satisfies rigorously the partial 

 differential equations of Lame, equations which can be deduced 

 equally well from theories of energy as from molecular hypotheses. 

 The molecular theory has therefore been a valuable guide for the 

 analyst m suggesting the course to be followed in studying the 

 equations of the problem, but it is eliminated from the ultimate 

 solution. On the other hand, we know that this solution represents 

 reality only imperfectly. We obtain an mfinitude of proper periods, 

 mstead of a very great number of them; the actual number so great, 

 mdeed, that one perhaps ought not to scruple to pass to the limit and 

 to regard it as practically mfinite. If, however, one observes that the 

 difficulties of the theory of black radiation come precisely from the 

 very short periods, and that these difficulties are not yet resolved in 

 an entirely satisfactory manner, one will perhaps judge that he can 

 not be too careful in ever>'^thmg which concerns these very short 

 periods. Perhaps this is why a physicist like Lorentz has not deemed 

 superfluous the considerable analytical efforts which the study of 

 the propagation of waves requires when we explicitly introduce the 

 molecules. However it may be in other cases, even if the substitu- 

 tion of the infinite for the finite is entirely legitimate in certain 

 problems, it may be interesting to propose to oneself, from a purely 

 mathematical point of view, the direct study of functions or equa- 

 tions depending upon a great, but fuiite number of variables. 



IV. 



The first difficulty which presents itself, when one wishes to study 

 functions of a great number of variables, is the exact definition of 

 such a function. I mean by that an individual definition, permittmg 

 one to distinguish the definite function from an infuiitude of other 

 analogous functions. There exist many general properties common 

 to the mathematical entities of a certain categoiy, mdependent of 

 the numerical value of the coefficients; for example, every defuiite 

 quadratic form (that is to say, one always positive) is equal to the 

 sum of the squares of as many mdependent luiear functions as the 

 number of the variables which it contams. One has at times sought 

 to deduce mathematical facts from this sort of physical consequences. 

 I must confess that I can not defend myself against some suspicion 

 in regard to this sort of reasoning; for it appears a little singular that 



