On pedal tetrahedra

P 1 , = -o (rq — sp), p which gives the result required. Since my Solution 2 was, quite unintentionally, rather unfair to the method of partial derivatives, I feel that I ought to draw attention to this shorter solution. The fact that the above solution is merely shorter than the one which I gave does not however detract from the practical advantages of the differential method. Any experienced teacher knows that the step which presents real difficulty to the beginner is the obtaining of equation (1) above. Although in the case of the example which I happened to choose for illustration (and it may not have been the best for the purpose) the above solution by partial derivatives happens to be quite as short as the solution by differentials, the fact remains that, while the technique of differentiation, when once understood, is almost " fool-proof," the pitfalls for the beginner in the solution given above are well known to every teacher of the subject. While the solution of a problem by partial derivatives may be quite a difficult piece of manipulation, exactly the same technique is required for the solution of a problem by differentials, however simple or complicated the problem in question may happen to be.