When an expression has to be integrated which contains the square root of a rational function of the first or second degree, the integral can be expressed in terms of the ordinary algebraic functions or the elementary transcendental functions, viz., exponential and circular. But when the polynomial under the radical is higher than the second degree its integral in general can only be expressed by means of transcendentals of a higher kind. The particular case in which the expression under the square root is a cubic or quartic gives rise to a class of definite integrals called Elliptic Integrals, because by means of them, as we shall see, we can express the length of the arc of an ellipse or other central conic.

To divide the straight line AB (containing a units) at C so that

By algebra, taking the positive root,

The number p may therefore have any positive value, integral or fractional, and when negative cannot exceed ¼. Secondly, AC and AB are incommensurable except when 4p +1 is a square:– e.g., if 4p = (q – l)(q +1) or if p = q(q+l), q being any positive integer or fraction.

The condition that the normals at the points whose eccentric angles are α, β, γ shall be concurrent is

The following method of establishing this result, as compared with those given in works on Concis, is direct, and also has the advantage of simplicity, by first proving the Trigonometrical identity

which may be simply done by multiplying both sides of the wellknown identity

when it will be easily found that the product in the first member reduces to that in (A) with opposite sign.

Theorem: If any conic be inscribed in a given triangle and a confocal to it pass through the circumcentre, then the circle through the intersection of these two confocals touches the nine-points circle of the triangle.

The possibility of the steady motion of a spherical vortex of constant vorticity in an infinite homogeneous liquid was first pointed out by Hill in the Phil. Trans., 1894, pp. 213–245. He had already discussed a case of motion which had for the surfaces always containing the same particles those given by the equation

a particular surface being

The following proof of this theorem assumes only Euclid, I. 43, and its converse, with the well-known deductions, “the line joining the mid points of two sides of a triangle is parallel to the third side,” and “the mid point of one diagonal of a parallelogram is also the mid point of the other.” The proof given by Dr Taylor in his Conies which suggested the method, makes use of ratios.

The following proof of the fundamental Combination Theorem does not appear in any of the current text-books on Algebra. It has the twofold advantage of being exceedingly simple and of being quite independent of the fundamental Permutation Theorem.

The object of this note was to point out that in using the method of limits to find a geometrical maximum or minimum it is not correct to conduct all the reasoning at the final stage when the limit has been reached, and to call attention to the form of statement which lays stress on the fact that the reasoning should be based on the consideration of the quantities involved while they are yet finite. Examples were given from one or two well-known books for students where the fallacious method of proof is adopted.

Direct geometical proofs of the addition theorem, and others allied to it, are often valid for only a limited range of values, and the constructions used are applicable to only one set of formulaæ. In the following paper a method is given which is valid for angles of all values and which is applicable to all the usual formulae involving sines, cosines, and their simple products. One of each set is proved: the others may be obtained either directly or by substitution in the usual manner.

In this paper the following expansion will be obtained:

in which n and r are positive integers. The Series in the square brackets are Hypergeometric Series with a finite number of terms.

On the sides BC, CA, AB of the triangle ABC are described

two sets of equilateral triangles,

the set Ba′C, Cb′A, Ac′B externally, and

the set BaC, CbA, AcB internally.

The lines Aa′, Bb′, Cc′ cointersect in Q, the centre of Perspective

of the triangles ABC, a′b′c′,

and the lines Aa, Bb, Cc in P, the centre of Perspective of ABC, abc.

Since a, a′, b, b′, c, c′ are on the perpendicular bisectors of BC, CA, AB, their joins cointersect in the circumcentre, O, which is the centre of Perspective of abc, a′b′c′.

If on the sides of a triangle ABC equilateral triangles LBC MCA NAB be described externally, AL BM CN are equal and concurrent.