A Philosophical Essay on Probabilities

by marquis de Pierre Simon Laplace

About the same time Moivre was considering under the name of recurring series the equations of finite differences of a certain order having a constant coefficient. He succeeded in integrating them in a very ingenious manner. As it is always interesting to follow the progress of inventors, I shall expound the method of Moivre by applying it to a recurring series whose relation among three consecutive terms is given. First he considers the relation among the consecutive terms of a geometrical progression or the equation of two terms which expresses it. Referring it to terms less than unity, he multiplies it in this state by a constant factor and subtracts the product from the first equation. Thus he obtains an equation among three consecutive terms of the geometrical progression. Moivre considers next a second progression whose ratio of terms is the same factor which he has just used. He diminishes similarly by unity the index of the terms of the equation of this new progression. In this condition he multiplies it by the ratio of the terms of the first progression, and he subtracts the product from the equation of the second progression, which gives him among three consecutive terms of this progression a relation entirely similar to that which he has found for the first progression. Then he observes that if one adds term by term the two progressions, the same ratio exists among any three of these consecutive terms. He compares the coefficients of this ratio to those of the relation of the terms of the proposed recurrent series, and he finds for determining the ratios of the two geometrical progressions an equation of the second degree, whose roots are these ratios. Thus Moivre decomposes the recurrent series into two geometrical progressions, each multiplied by an arbitrary constant which he determines by means of the first two terms of the recurrent series. This ingenious process is in fact the one that d'Alembert has since employed for the integration of linear equations of infinitely small differences with constant coefficients, and Lagrange has transformed into similar equations of finite differences.

Finally, I have considered the linear equations of partial finite differences, first under the name of recurro-recurrent series and afterwards under their own name. The most general and simplest manner of integrating all these equations appears to me that which I have based upon the consideration of discriminant functions, the idea of which is here given.

If we conceive a function V of a variable t developed according to the powers of this variable, the coefficient of any one of these powers will be a function of the exponent or index of this power, which index I shall call x. V is what I call the discriminant function of this coefficient or of the function of the index.

Now if we multiply the series of the development of V by a function of the same variable, such, for example, as unity plus two times this variable, the product will be a new discriminant function in which the coefficient of the power x of the variable t will be equal to the coefficient of the same power in V plus twice the coefficient of the power less unity. Thus the function of the index x in the product will be equal to the function of the index x in V plus twice the same function in which the index is diminished by unity. This function of the index x is thus a derivative of the function of the same index in the development of V, a function which I shall call the primitive function of the index. Let us designate the derivative function by the letter Alembert placed before the primitive function. The derivation indicated by this letter will depend upon the multiplier of V, which we will call T and which we will suppose developed like V by the ratio to the powers of the variable t. If we multiply anew by T the product of V by T, which is equivalent to multiplying V by T², we shall form a third discriminant function, in which the coefficient of the xth power of t will be a derivative similar to the corresponding coefficient of the preceding product; it may be expressed by the same character δ placed before the preceding derivative, and then this character will be written twice before the primitive function of x. But in place of writing it thus twice we give it 2 for an exponent.

Continuing thus, we see generally that if we multiply V by the nth power of T, we shall have the coefficient of the xth power of t in the product of V by the nth power of T by placing before the primitive function the character δ with n for an exponent.

Let us suppose, for example, that T be unity divided by t; then in the product of V by T the coefficient of the xth power of t will be the coefficient of the power greater by unity in V; this coefficient in the product of V by the nth power of T will then be the primitive function in which x is augmented by n units.

Let us consider now a new function Z of t, developed like V and T according to the powers of t; let us designate by the character Δ placed before the primitive function the coefficient of the xth power of t in the product of V by Z; this coefficient in the product of V by the nth power of Z will be expressed by the character Δ affected by the exponent n and placed before the primitive function of x.

If, for example, Z is equal to unity divided by t less one, the coefficient of the xth power of t in the product of V by Z will be the coefficient of the x + 1 power of t in V less the coefficient of the xth power. It will be then the finite difference of the primitive function of the index x. Then the character Δ indicates a finite difference of the primitive function in the case where the index varies by unity; and the nth power of this character placed before the primitive function will indicate the finite nth difference of this function. If we suppose that T be unity divided by t, we shall have T equal to the binomial Z + 1. The product of V by the nth power of T will then be equal to the product of V by the nth power of the binomial Z + 1. Developing this power in the ratio of the powers of Z, the product of V by the various terms of this development will be the discriminant functions of these same terms in which we substitute in place of the powers of Z the corresponding finite differences of the primitive function of the index.

Now the product of V by the nth power of T is the primitive function in which the index x is augmented by n units; repassing from the discriminant functions to their coefficients, we shall have this primitive function thus augmented equal to the development of the nth power of the binomial Z + 1, provided that in this development we substitute in place of the powers of Z the corresponding differences of the primitive function and that we multiply the independent term of these powers by the primitive function. We shall thus obtain the primitive function whose index is augmented by any number n by means of its differences.

Supposing that T and Z always have the preceding values, we shall have Z equal to the binomial T - 1; the product of V by the nth power of Z will then be equal to the product of V by the development of the nth power of the binomial T - 1. Repassing from the discriminant functions to their coefficients as has just been done, we shall have the nth difference of the primitive function expressed by the development of the nth power of the binomial T - 1, in which we substitute for the powers of T this same function whose index is augmented by the exponent of the power, and for the independent term of t, which is unity, the primitive function, which gives this difference by means of the consecutive terms of this function.