NATURE 725 



variable, but varies continuously. We shall then agree to say that 

 the probable number of representative points to be found on a certain 

 portion of the plane is proportional to the quantity of this imaginary 

 matter which is found there. If there are, then, two regions of the 

 plane of the same extent, the probabilities that a representative point 

 of one of our minor planets is in one or other of these regions will be 

 as the mean densities of the imaginary matter in one or other of the re- 

 gions. Here then are two distributions, one real, in which the represent- 

 ative points are very numerous, very close together, but discrete like 

 molecules of matter in atomic hypothesis ; the other remote from reality 

 in which our representative points are replaced by imaginary continuous 

 matter. We know that the latter cannot be real, but we are forced 

 to adopt it through our ignorance. If, again, we had some idea of the 

 real distribution of the representative points, we could arrange it so 

 that in a region of some extent the density of this imaginary continu- 

 ous matter may be nearly proportional to the number of representa- 

 tive points, or if it is preferred, to the number of atoms which are 

 contained in that region. Even that is impossible, and our ignorance 

 is so great that we are forced to choose arbitrarily the function which 

 defines the density of our imaginary matter. We shall be compelled 

 to adopt a hypothesis from which we can hardly get away; we shall 

 suppose that this function is continuous. That is sufficient, as we 

 shall see, to enable us to reach our conclusion. 



What is at the instant t the probable distribution of the minor 

 planets or rather, what is the mean value of the sine of the longi- 

 tude at the moment i i.e., of sin (at+b) ? We made at the outset 

 an arbitrary convention, but if we adopt it, this probable value is en- 

 tirely denned. Let us decompose the plane into elements of surface. 

 Consider the value of sin (ai+b) at the centre of each of these ele- 

 ments. Multiply this value by the surface of the element and by the 

 corresponding density of the imaginary matter. Let us then take the 

 sum for all the elements of the plane. This sum, by definition, will 

 be the probable mean value we seek, which will thus be expressed by a 

 double integral. It may be thought at first that this mean value 

 depends on the choice of the function </> which defines the density of 

 the imaginary matter, and as this function <f> is arbitrary, we can, 

 according to the arbitrary choice which we make, obtain a certain 

 mean value. But this is not the case. A simple calculation shows us 

 that our double integral decreases very rapidly as t increases. Thus, 

 I cannot tell what hypothesis to make as to the probability of this or 

 that initial distribution, but when once the hypothesis is made the 

 result will be the same, and this gets me out of my difficulty. What- 

 ever the function < may be, the mean value tends towards zero as t 

 increases, and as the minor planets have certainly accomplished a 

 very large number of revolutions, I may assert that this mean value is 



