The determination of the errors involved in interpolation or quadrature formulae often involves complicated reasoning. There are many cases, however, where the remainder can be obtained very easily. These cases belong to the class which we call simplex.

In general a formula will be expressible in the form

L(f) =∑rArf(xr) + ∑sBsf'(xs)+ ∑tCtf''(xt) +… ......(1)

to a finite number of terms and it is assumed throughout that f(x) possesses derivatives as far as the remainder may require.

Introduction to the principles of elasticity is often received by the average student in a rather piecemeal fashion, the ideas of strain, shear, etc., being indicated from consideration of a number of very special cases. There is a danger that the student may not see the forest for the trees for a considerable time, and a more general approach to the analysis of strain, if not too difficult, would have obvious advantages. The general presentation as in Love's Elasticity is probably too lengthy for a first approach to the subject, and the following method is suggested as being perhaps suitable for the purpose. Any difficulty that a junior student might find with the mathematics used is compensated by the extreme conciseness of the method. Moreover, the mathematical methods used are all important ones, and the treatment might well serve also as a first introduction to some of these methods. The incidental derivation of the rotation formula would have the further advantage that the student would be shown at an early stage the significance of the curl of a vector.

It sometimes happens that teachers want missing figure problems and coded sums, for instance, at the end of term. These given here may be of use in saving teachers from having to make them up themselves. To save space, the missing figure problems are given as a whole, with the data printed in clarendon type. In setting these problems to a class the exact position of every missing digit must, of course, be indicated : this applies to remainders as well as to the main sum.

Here again we are confronted with a variety of methods. For purposes of work in the early stages at least two of these are worthy of consideration. They are the factorial or Ten-ten and the whole-sale methods. This latter is sometimes called the “ Ballard Box Method “ although it was known and used (but not extensively) long before Dr. Ballard's work on this subject.

It is generally accepted among teachers that there are certain forms of arrangement in arithmetic with which pupils are more comfortable and more sure. Addition, for example, is usually considered to be more rapid and more accurate when the addenda are placed beneath each other as in vertical tots than when they are placed side by side as in across tots.

When considering the advantages and disadvantages of the different methods used in long multiplication of money I decided to commence with an investigation into the various forms of setting out short multiplication of money. The surprising results obtained led me to extend the work to other items of the arithmetic syllabus. Each item of the whole investigation is recorded in the chronological order with which it was dealt.

There are two forms of setting out questions involving the reduction of shillings to £'s. They are:

In example (i), after crossing off the nought of the 20 and the last figure of the dividend, division is by 2 ; whilst in example (ii) division is by 20. Consequently, in (ii), the first figure of the quotient is one place further to the right than it is in (i). The divisor, 20, is often omitted in both examples.