the first two players who has beaten his adversary should beat at the second play the third player; the probability that the game will end at this play is ½. Hence by virtue of the preceding equation we conclude that the successive probabilities of the end of the game are ¼ for the third play, ⅛ for the fourth play, and so on; and in general ½ raised to the power n - 1 for the nth play. The sum of all these powers of ½ is unity less the last of these powers; it is the probability that the game will end at the latest in n plays.
Let us consider again the first problem more difficult which may be solved by probabilities and which Pascal proposed to Fermat to solve. Two players, A and B, of equal skill play together on the conditions that the one who first shall beat the other a given number of times shall win the game and shall take the sum of the stakes at the game; after some throws the players agree to quit without having finished the game: we ask in what manner the sum ought to be divided between them. It is evident that the parts ought to be proportional to the respective probabilities of winning the game. The question is reduced then to the determination of these probabilities. They depend evidently upon the number of points which each player lacks of having attained the given number. Hence the probability of A is a function of the two numbers which we will call indices. If the two players should agree to play one throw more (an agreement which does not change their condition, provided that after this new throw the division is always made proportionally to the new probabilities of winning the game), then either A would win this throw and in that case the number of points which he lacks would be diminished by unity, or the player B would win it and in that case the number of points lacking to this last player would be less by unity. But the probability of each of these cases is ½; the function sought is then equal to one half of this function in which we diminish by unity the first
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 ½, ¾, 7⁄8, etc., according as B lacks one point, two, three, etc. Generally it is then unity less the power of ½, 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 ¼, ½, 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
they occupy in the series. This equation I call the equation of finite differences by a single index. The order or the degree of this equation is the difference of rank of its two extreme terms. We are able by its use to determine successively the terms of the series and to continue it indefinitely; but for that it is necessary to know a number of terms of the series equal to the degree of the equation. These terms are the arbitrary constants of the expression of the general term of the series or of the integral of the equation of differences.