The following paper was written last summer, and was submitted to Dr Mackay with a view to eliciting his opinion particularly on the curious passage referred to in § 3, and on the remarks contained in § 8. I was not aware of the intention of Mr T. L. Heath to follow up his excellent edition of Apollonius by an edition of Archimedes on similar lines, and when I saw the announcement of his Archimedes in the month of October, I at once concluded that the notes I had made would have been anticipated by him. Since reading his masterly work, however, I am disposed to think there is still sufficient interest in the notes I have written to justify me in laying them before the Society; I therefore submit them in their original form, although I should have omitted certain details had I been acquainted with Mr Heath's work before writing the paper.

On est amené à résoudre l'équation aux différences mêlées

lorsqu'on cherche, en coordonnées rectangulaires, une courbe telle que: M et M′ étant deux de ses points dont les abscisses différent d'une constante, la tangente de l'angle que fait avec OX la tangente en M soit en raison inverse de la sous-tangente en M′.

This summer there came into my hands a copy of the spring issue of the Mittheilungen der Math. Gesellschaft in Hamburg containing a paper on the “Transformationen der hydrodynamischen Gleichungen mit Berucksichtigung der Reibung.”

On examination, I found embodied in the somewhat lengthy communication practically the following method, which I had entered in my notes three years ago when working at the subject. Thinking that it was bound to have been used earlier, I simply preserved it, as likely to prove useful if I were ever called upon to teach Hydrodynamics.

It is assumed that the sines of the sides are proportional to the sines of the opposite angles.

ACB (Fig. 9) is a spherical Δ, with C a right angle.

Produce AC, AB to D and E so that

Draw the great ⊙ DEF and produce CB to meet it in F.

Then F is the pole of AD and A the pole of DF.

Now a third Δ can similarly be derived from this second, a fourth from the third, and a fifth from the fourth. But when the process is applied to the fifth, the first Δ is obtained. Hence only 5 Δs can be obtained, which are the following:-

where the mid-column contains the hypotenuse, the two next to it contain the angles, and the extreme columns the sides of the several right-angled triangles.

In a given circle let the arc AP subtend an angle 3a at the centre O, it is required to trisect the angle AOP, or the arc AP.

The three trisectors will be OQ1, OQ2, OQ3, where AOQ1 = a,

∴ Q1 Q2 Q3 form an equilateral triangle. (See Figs. 10 and 11.)

We proceed to solve the problem by drawing a conic through Q1, Q2, Q3, a nd we wish to find in what cases such a conic can be drawn, a conic cutting the circle in four points, three of which form an equilateral triangle.

To find the Centre of Gravity of a Ciroular Arc.

Let a (Fig. 12) be the radius, 2a the angle at the centre, AB the arc, of total mass m, G its centre of gravity symmetrically situated.

Imagine the arc to be part of a circle of string rotating uniformly with velocity u round C and of linear density ρ

Stewart, a Scotch geometrician, gave in 1763 the demonstration of the following theorem: “If we divide the base of a triangle into two segments by a straight line going through the vertex, the sum of the squares of the two sides, multiplied each by the non-adjacent segment, is equal to the product of the base multiplied by the square of the straight line plus the rectangle contained by the two segments.”

Mr Crawford's note on this subject, read at a recent meeting, reminds me of a method I gave to a class four or five years ago.

The following two modes of generation of the wave-surface are pretty generally known.

(a) A given ellipsoid (surface of elasticity) is cut by any central plane π along an ellipse of semiaxes λ1 and λ2. If π varies, the two pairs of planes π∲ parallel to π at distances k2/λ1, k2/λ2 (k = constant) from it envelope the wave-surface W1 represented by the tangential equation

if the tangential coordinates u1 depend on Σux +1 = 0.

I consider here the second solutions corresponding to the solutions of the above equation (when n is an integer) in finite terms for special values of B. If Un be such a solution, and Fn the corresponding second solution, we know that

Many theories have been advanced to account for a presumed uniformity in the temperature of the earth in past ages, and one of the most recent is that advanced by Sir John Murray in the Summary Volumes of the Challenger Report. A careful study of the distribution of marine fauna showed the existence of remarkably similar organic forms in the Arctic and Antarctic regions which were entirely unknown in intermediate waters. The existence, also, of ancient corals in Polar seas points to a high Polar temperature at some remote period of the earth's history. These facts combined lead us to consider it probable that in some past age the earth's air temperature was high and uniform.

The subject of the Singular Solutions of Differential Equations of higher orders than the first is not touched in the ordinary textbooks. Their existence, for instance, is not mentioned by Forsyth in his Treatise. This is probably due to the fact that, while in the case of equations of the first order a theory has been developed by Cayley and others which connects the singular solution in a geometrical manner with the ordinary solutions (the singular solution being, of course, the envelope of the ordinary solutions), in the case of equations of, say, the second order no corresponding theory exists—at any rate, no corresponding theory has yet been developed. Our only guide in the subject at present is Cauchy's Existence Theorem, which points out where we are to look for singular solutions.

The number of the r-permutations of n things is the same as the number of ways of adding r things to a row which originally contains n - r other things.

For suppose the n letters A, B, C … N to be placed in a row, then each permutation consisting of r letters may be indicated by placing below those r letters in the row, the digits 1, 2, 3 … r, to indicate the order they have in the permutation. We may suppose zeros placed below all the remaining n - r letters. Thus it is clear that the number of the r-permutations of n things is the same as the number of ways in which r numbers can be added to a row originally containing n - r zeros.