Suppose in an urn _a_ white balls, _b_ black balls, and after having drawn a ball it is put back into the urn; the probability is asked that in _n_ number of draws _m_ white balls and _n_ - _m_ black balls will be drawn. It is clear that the number of cases that may occur at each drawing is _a_ + _b_. Each case of the second drawing being able to combine with all the cases of the first, the number of possible cases in two drawings is the square of the binomial _a_ + _b_. In the development of this square, the square of a expresses the number of cases in which a white ball is twice drawn, the double product of _a_ by _b_ expresses the number of cases in which a white ball and a black ball are drawn. Finally, the square of _b_ expresses the number of cases in which two black balls are drawn. Continuing thus, we see generally that the _n_th power of the binomial _a_ + _b_ expresses the number of all the cases possible in _n_ draws; and that in the development of this power the term multiplied by the _m_th power of _a_ expresses the number of cases in which _m_ white balls and _n_ - _m_ black balls may be drawn. Dividing then this term by the entire power of the binomial, we shall have the probability of drawing _m_ white balls and _n_ - _m_ black balls. The ratio of the numbers _a_ and _a_ + _b_ being the probability of drawing one white ball at one draw; and the ratio of the numbers _b_ and _a_ + _b_ being the probability of drawing one black ball; if we call these probabilities _p_ and _q_, the probability of drawing _m_ white balls in _n_ draws will be the term multiplied by the _m_th power of _p_ in the development of the _n_th power of the binomial _p_ + _q_; we may see that the sum _p_ + _q_ is unity. This remarkable property of the binomial is very useful in the theory of probabilities. But the most general and direct method of resolving questions of probability consists in making them depend upon equations of differences. Comparing the successive conditions of the function which expresses the probability when we increase the variables by their respective differences, the proposed question often furnishes a very simple proportion between the conditions. This proportion is what is called _equation of ordinary or partial differentials_; _ordinary_ when there is only one variable, _partial_ when there are several. Let us consider some examples of this.