When the subject of the teaching of differentials was discussed by this Association at its Annual Meeting two years ago, I read a paper on the theoretical aspect of the subject. In that paper I maintained that it is futile to discuss the teaching of differentials until we have agreed upon the theory of differentials which we wish to teach, and I proceeded to summarise the three main theories of differentials. It was obvious that each of these theories was intrinsically more difficult and abstract than the theory of limiting processes which has become the classical method of introducing the calculus, and it was unanimously agreed that each theory was, in practice, quite unteachable.