MOLECULAR THEORIES AND MATHEMATICS BOREL. 173 



one can deduce anything exact from a notion so general as that of a 

 surface of the second degree (let us say, for fixing ideas, a generalized 

 ellipsoid) m a space havuig a very great number of dimensions. Let 

 us emphasize a little the difficulty that there is m kno\\ing individually 

 such an ellipsoid : Its equation may be supposed to be reduced to the 

 sum of squares, by an orthogonal substitution; that is to say, the axes 

 remaming rectangular. Such an ellipsoid, then, requkes for its com- 

 plete definition the laiowledge of what we may call the squares of 

 the lengths of its axes; that is to say, the squares of as many positive 

 numbers as the space considered contains dimensions. The question 

 of knowing whether one can consider as given so many numbers, when 

 a man's lifetune would not suffice to enumerate a small })art of them, 

 is a question which is not without analogy to that of the legitunacy of 

 certam reasonings m the theory of ensembles, such as that one by 

 which Prof. Zermelo pretends to prove that the continuum can be well 

 ordered, and which suppose, as realized, an mffiiitude of choices inde- 

 pendent of all law, and at the same time to be uniquely determined. 

 Opinions may differ on the theoretical solution of these difficulties 

 and here is not the place to reopen this controversy. But, from the 

 practical pomt of view, the response is not doubtful; it is not possible 

 to actually \\Tite the numerical equation of an ellipsoid whose axes 

 are as numerous as the molecules constituting a gram of hydrogen. 



In what sense is it, then, possible to speak of a numerically deter- 

 minate ellipsoid, possessing a very great number of dimensions? 

 From an abstract j^oint of \T.ew, the most shnple process for defining 

 such an ellipsoid, consists in supposing that the lengths of the axes 

 are equal to the values of a certain function which is simple for the 

 integral values of the variable. One can suppose them aU equal (iu 

 which case he will say that the ellipsoid is reduced to a sphere). 

 One can also supj)ose that they have for values the successive integral 

 numbers taken in their natural order, starting either with unity or 

 \\ith any other given number, or that they are equal to the inverses 

 of the square of these integers, etc. In other words, we suppose that 

 the lengths of the axes arc all determined by the Imowlcdge of a 

 formula simple enough for being actually written, while it is not 

 possible to actually write as many distinct numbers as there arc axes. 



iVnother process, to which we arc natui'ally led by the analogies 

 with the kinetic theory of gases, consists in supposing that the values 

 of a function of the axes such that the square of the lengths of the 

 axes, or of their reciprocals, etc., are not given individually, but that 

 we know only the mean value of this function, and the law of the 

 distributions of the other values around tlds mean. We propose, 

 under those conditions, not to study the properties of a unique and 

 well determined ellipsoid, but only the most probable properties of the 



