Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/42

32 index plus the half of the same function in which the second variable is diminished by unity. This equality is one of those equations called equations of partial differentials.

We are able to determine by its use the probabilities of A by dividing the smallest numbers, and by observing that the probability or the function which expresses it is equal to unity when the player A does not lack a single point, or when the first index is zero, and that this function becomes zero with the second index. Supposing thus that the player A lacks only one point, we find that his probability is $1⁄2$, $3⁄4$, $7⁄8$, etc., according as B lacks one point, two, three, etc. Generally it is then unity less the power of $1⁄2$, equal to the number of points which B lacks. We will suppose then that the player A lacks two points and his probability will be found equal to $1⁄4$, $1⁄2$, $11⁄16$, etc., according as B lacks one point, two points, three points, etc. We will suppose again that the player A lacks three points, and so on.

This manner of obtaining the successive values of a quantity by means of its equation of differences is long and laborious. The geometricians have sought methods to obtain the general function of indices that satisfies this equation, so that for any particular case we need only to substitute in this function the corresponding values of the indices. Let us consider this subject in a general way. For this purpose let us conceive a series of terms arranged along a horizontal line so that each of them is derived from the preceding one according to a given law. Let us suppose this law expressed by an equation among several consecutive terms and their index, or the number which indicates the rank that