The condition in question may be found by using the following theorem:-

If the equation in trilinear co-ordinates

The problem “to inflect a straight line between two sides of a triangle so that the intercepted portion is equal to the segments cut off” has been discussed in the third volume of the Proceedings.

If we discuss the same analytically; taking CB and CA as axes of x and y (Fig. 1) and calling each segment k, the equation of the line considered is

The greatest line joining two points in the perimeter of a polygon is a side or a diagonal of the polygon.

For (fig. 2) PS is obviously less than one or other of the lines PQ, PR. Hence if AB, CD (fig. 3) are any two finite lines, L and M any points, one in each of these lines, LM is less than LC, or LD and LC is less than CA or CB. From this the theorem follows at once. The theorem and the above proof apply to crossed and gauche polygons.

If P=a+b−c−d, and K=cd−ab, the solution of the equation

may be most readily found by putting (x−a)(x−b)=z2; whence (x−c)(x−d)=z2+Px+K, and therefore =z2−2ez+e2, and finally x is given by the equation

Let it be required to find the square integral numbers which added to a given integer shall produce a square integer, and the smallest such number.

Par un point fixe A d'une circonférence donnée on mène deux cordes AB et AC dont le produit a une valeur constante m2, puis on joint BC. Trouver 1° le lieu du pied D de la bissectrice de l'angle A du triangle ABC; 2° le lieu des centres des cercles inscrits et exinscrits à ce triangle.

The perpendicular from A on BC = AB·AC/2R, which is constant.

In what follows, a triangle ABC is taken, from it another triangle A′B′C′ is formed so that

A is the mid point of B′C′,

B is the mid point of C′A′,

and C is the mid point of A′B′.

Though every quantity, whatever be its nature, has magnitude, no quantity can be said to be large or small absolutely. When we speak of the size of any body we mean its size relatively to the size of some other body with which we compare it. A yard is large if we compare it with an inch; it is small when compared with a mile. In the former case the number which represents it is more than 60,000 times larger than the number by which it is represented in the latter case. A mere number is therefore useless as regards the statement of magnitude, except when accompanied by a clear indication of what the thing measured is compared with. The quantity in terms of which the comparison is made is called the unit, and the number which tells how often this unit is contained in a given quantity is called the numeric of that quantity.

Let ABC be the triangle, X, Y, Z the feet of the altitudes, H the orthocentre, A′,B′,C′ the mid points of the sides.

Let ZY meet A′B′ in D, A′C′ in D′, B′C′ in R:

ZX meet B′C′ in E, A′B′ in E′, A′C′ in S:

XY meet C′A′ in F, B′C′ in F′, A′B′ in T.

It is well known that in a given triangle a one-fold infinity of triangles may be inscribed similar to a given triangle This becomes at once obvious on consideration of the converse problem; for we may circumscribe about a given triangle (A), a triangle similar to a second triangle (B), and having its sides parallel to the sides of (B).

1. Taking the two following known properties of the pedal line of a triangle, viz.:

I. The locus of a point, such that the feet of the perpendiculars from, it on the sides of a triangle are collinear, is the circum-circle of the triangle;

II. The pedal line bisects the distance between the orthocentre and the corresponding point in the circumference of the circum-circle;

Most of the following theorems occur in a more or less explicit form in text-books on the geometry of the parabola; but it will not, I hope, be without interest and value to consider them independently, and to prove them by using only the propositions of Euclid.

It is well known that the elliptic integral of the second kind may be represented by the arc of an ellipse, and mathematicians have sought with various success to represent similarly by the arc of an algebraic curve the elliptic integral of the first kind. The general solution of the problem has not been obtained, but Serret and Cayley have given solutions of a very general character.

Let represent any permutation of the first n natural numbers. If we take the first number, a1, and put it last, we get a new permutation; and if we perform this operation (n−1) times, we get the (n−1) permutations