174 ANNUAL EEPOET SMITHSONIAN INSTITUTION, 1912. 



ellipsoid, lalo^villg only that it satisfies the imposed conditions. We 

 can also say that we study the mean properties of the ensemble of 

 the ellipsoids defined by these conditions. Here again, we may ob- 

 serve that the probable ellipsoid or the mean ellipsoid is completely 

 defined by the knowledge of the mean value of the law of the devia- 

 tions. If tills law is the classic law of probabilities, it includes only 

 two constants. If we were led to introduce a more complicated law, 

 this law might in aU cases be explicitly written. The two processes 

 that we have indicated are then equivalent from the analytical point 

 of view. It would evidently be the same with all other processes 

 that we can imagme, and notably with the combmations of the two. 



In a word, a figure which depends on an extremely great number of 

 parameters can be considered a numerically determinate only if these 

 parameters are defined by means of numerical data sufficiently few 

 in number to be accessible to us. It is for this reason that the study 

 of the geometrical figures in a space possessmg an extremely great 

 number of dimensions, can lead to general laws, if we can exclude 

 from this study such of these figures as it is impossible for a human 

 being to define individually. 



Here are, for example, some of the results to which one is led by 

 the study of elHpsoids. In writing the equations in the form of a 

 sum of squares, the second member bemg reduced to unity, the co- 

 efficients are equal to the reciprocals of the squares of the axes. If 

 the mean of the squares of these coefficients is of the same order of 

 magnitude as the square of their mean, one will say that the ellipsoid 

 is not very ii-regular. The modes of defhiition concerning which we 

 have just spoken lead to 'ellipsoids which are not very ii-regular, from 

 the moment when one ceases to systematically introduce into these 

 definitions functions purposely chosen in a complicated manner. On 

 the other hand, we obtain a very UTegular ellipsoid in equating to a 

 constant the vis viva of a deformable system composed of a very great 

 number of molecules, this vis viva being writen under the classic 

 form of the sum of the vis viva of translation of the total mass con- 

 centi'ated at the center of gravity, increased by the sum of the vires 

 vivse of the molecules in their motion relative to this center of gravity. 

 The great ii-regularity comes from the fact that the products of the 

 total mass by the thi-ee components of the velocity of the center of 

 gravity are extremely great in comparison with the other terms. 

 Wlien an ellipsoid is not very irregular, several of its properties permit 

 comparing it to a sphere, which we may call the median sphere. 

 The surface of the ellipsoid is ahnost wholly comprised between the 

 surfaces of two spheres very near to the median sphere. On the 

 other hand, a pomt being arbitrarily chosen on the ellipsoid, it is 

 infinitely probable that the normal at tliis point passes extremely 

 close to the center. 



