Mr Ridley's note in the January number giving an algebraic explanation of extraneous values suggests an elementary treatment of the subject on the geometrical side. It must be assumed that the pupil is able to trace from their equations straight lines, circles, and parabolas of the type y = ax2 + bx + c. He should be made to draw the curves in cases like those given here, and to state the geometrical meaning of every distinct step in the algebra.

Consider the curve y = ae−t/t0 (Fig. 1.)

Mark off along the t – axis a distance OT0 equal to the time constant t0.

Divide OT0 into n equal parts and erect ordinates at each point of division.

M, N, are the mth and (m + 1)th of these, counting from O.

MP, NQ, are the corresponding ordinates; P, Q, the points on the curve; and R the intersection of the chord, produced, with the t – axis.

Integrate tanx; by parts as follows:—

i.e. 0= — 1, which is absurd.

It might be conjectured at first sight that the above result is due to a disregarded constant of integration, but this is not so, because we can imagine all the integrals taken between the same limits, in which case there is no constant. The true explanation is to be found by considering the proof of the theorem of integration by parts.

A velocity is a vector quantity, for it possesses magnitude and direction. It is, of course, necessary to make clear in every case what given point (moving or not) is considered as fixed for the time being, to which the vclocity of the moving point (so-called, though it may only be relatively moving) is referred.