Mr J. F. Cameron of Gonville and Caius College, Cambridge, has suggested the use of factorial functions such as or n! to solve Whewell's problem. I had excluded them as beyond the bounds of ordinary arithmetic. I append half a score of my attempts with the factorials; they could be easily increased.

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.

This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.

For n = 1, the theorem is that, if a = 0, the equation ax = 0 has a solution for which x is not zero. This is obviously true. And the case of n variables can easily be made to depend on that of n – 1 variables. For brevity we show the method by taking n = 3.

The following method for three variables can be extended at once to the general case.

Let z1, z2, z3, … be a, sequence of quantities, real or complex, such that, corresponding to any arbitrary small positive quantity ∈, we can find a positive integer n such that | zn+p – zn | < ∈ for all positive integral values of p.