§1. Descartes in his Geometria shows how the solution may be made to depend on the intersections of two conies C and C′; an account of this method is given in the Algebra of Maclaurin. Thus the roots of x4 + qx2 + rx + s = 0 are the x-coordinates of the points common to the parabola y = x2 and the circle x2 + Y2 + rx + (q – 1) y + s = 0.

ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy = 1.

In Mathematical Notes, No. 20 (April 1916), there is a note on the above; I add a form of the equations of these axes which I have not seen in a text-book, and which is perhaps worth recording.

In his article in No. 20, Professor Gibson gives proofs of the inequalities with certain restrictions as to the values of n and a. The mode of proof, avoiding as it does the use of the Binomial Theorem, is comparatively short and simple. The following mode of proof is, I think, still simpler, as it does not involve the use of Mathematical Induction.

Let AB and BC be two circular arcs subtending angles 2α and 2β at the common centre 0. From symmetry the centroids G1, G2 and G of AB, BC and AC lie on the bisectors OP, OQ and

OR of the angles which they subtend at the centre. Also, G is the centroid of two particles placed at G1 and G2, and with masses proportional to the arcs AB and BC. Hence G1, G, and G2 are collinear, and

Equating (1) and (2) we have

Hence the ratio is independent of α, and therefore

the angle α. being in circular measure.

The curve Y = x3 is constructed with a scale of 1″ horizontally and ″ vertically. It is graduated with the values of x. Then draw the two lines x = ± 1 and graduate them on the scale of the y-axis, x = + 1 positive downwards, x = – 1 positive upwards.

The following elementary methods of interpreting the equation of the first degree in two and three dimensions may be of some pedagogic interest.

Most continuously recording instruments are arranged so as to draw a graph exhibiting the quantity measured by means of cartesian coordinates. The record is on a roll of paper unwound from a drum. If the instrument be some form of meter, the value of the output, represented by ∫ y dx, may be obtained by measuring the area by a planimeter. The records, however, are somewhat bulky, and in this respect a polar graph is preferable. This is drawn by an indicator moving radially on a revolving disc. The peak or maximum of the curve is usually the most important, and this is the part which is represented best on such a polar diagram. Space may be further economised by having more than one convolution on one sheet of paper. Polar output records, however, are not freely used, perhaps because the planimeter does not give directly what is wanted, as it measures ½ ∫ r2-dθ, instead of ∫r dθ.