We turn now to the question of the generality of the solution of equation (2) obtainable by means of equation (3), that is, to the problem of the construction of integrals of the equation

In passing, we remark that it is shewn by the last example that integrals of equation (2) deduced from different functions satisfying the conditions of equation (3) may differ, so that a number of independent functions satisfying equations of the form of (3) may lead to an equivalent number of independent solutions of (2), or may lead to any smaller number other than zero.

If we seek approximations to in the form

of such a nature that the fraction gives, on performing the division, an infinite series whose first 2r+1 terms are identical with the first 2r+1 terms of the Binomial Expansion for we shall obtain 2r equations of the form