[The correspondence which is here printed was bought by me on the 28th of March 1887 at the sale of the Gibson-Craig collection of Scottish MSS.

Simson's letters, which are beautifully written, seem all to have passed through the post, but Stewart's letters are, I conjecture, merely the drafts of what he proposed to send. The handwriting of the latter, though legible, is not elegant, and there are frequent erasures. I have scrupulously respected, in all the letters, the spelling, the punctuation (or want of it), the use or disuse of capitals, and I have made no attempt to improve the style.

The use of Green's Functions in the Theory of Potential is well known. The function is most conveniently defined, for the closed surface S, as the potential which vanishes over S and is infinite as when r is zero, at the point P(x0, y0, z0), inside the surface. If this is represented by G(P), the solution with no infinity inside S and an arbitrary value V over the surface, is given by

denoting differentiation along the outward drawn normal.

The object of this paper is to investigate certain series and differential equations, generalizations of the series of Bessel and Legendre.

Throughout the paper

reducing, when p = 1, to n.

The series discussed are the following:—

a generalized form of Pn(x),

the relation between coefficients of x in successive terms being

In many problems of physics, even in widely different branches of the subject, the relation satisfied by the variables is expressible by means of a linear differential equation of the second order. In general, “initial” conditions have also to be satisfied. If the equation truly represents the physical conditions in, for example, some case of motion, and if no state of instability exists, the solution must be unique. But it is impossible in any case to say with absolute certainty that the representation is strict. The possible error depends on the error which may be made in observation or experiment, and on the number of independent observations or experiments the results of which have been used as the basis of the “law” expressed by the equation.

The tangent at a point P to a given plane curve intersects another given curve in Q and makes with the tangent at Q to the latter curve a variable angle ψ. It is required to connect the curvatures at P and Q with the length PQ and the angle ψ.

Definition. If x, y, z and ξ, η, ζ be the perpendiculars on the sides BC, CA, AB of the Δ ABC from points O and O′, then O and 0′ are antireciprocal points if xξ η, zζ:: tanA : tanA : tanB: tanC.

I. Construction to find a point antireciprocal to O (Fig. 4).

Draw through O a line MN antiparallel to BC. Draw OY perpendicular to AC, and OZ perpendicular to AB. Draw lines parallel to AB and AC, and at distances from them respectively equal to YN and MZ, and let them cut in P. Join AP. Find a similar line BQ, and let AP and BQ cut in O′.

In the general equation of a curve that passes through the origin,

u1 ≡ a1x + b1y = 0 is the equation to the tangent at the origin; for, y q g when x and y are very small, we may neglect all the terms in comparison with those involving the first powers of x and y. If neither a1 nor b1 vanishes, we may, without loss of generality, write the equation in the form

A method of expressing a sum of money as a decimal of a £ to 3 places has long been known. When further decimals are necessary, they may be got by the following simple rule, which appears to be new: multiply the last two decimals found at any stage of the process (after the first 3 have been obtained) by 4, take the digit in the ten's place of the product, append it to the decimals already found as the next decimal, and repeat the process; with the proviso that should the 4-product end with 48, 68, or 88 the digit in the ten's place is to be taken as 5, 7, or 9 respectively.

As stated in the preceding paper, the method of expressing, at sight, shillings, pence, and farthings as a decimal of a pound to 3 places has long been known. It is sometimes referred to as the actuaries' rule. According to De Morgan, it occurs for the first time in Kersey's edition of Wingate's Arithmetic, 1673 (p. 191). It is also to be found in Cocker's Decimal Arithmetic, 1685 (although in a form which is not quite accurate). In some of the earlier books the method of conversion at sight from the decimal form is given, but not vice versâ. It is now found in most modern text-books in one form or another.

The theorems given in the present section are fundamental in the theory of triangles in multiple perspective. They are all perfectly well known, but are given here because without them the succeeding sections would be unintelligible.

Two triangles A1,A2A3, B1,B2B3, can be in perspective in six different ways, indicated by the following symbols, in which the A's are to be understood as connecting collinearly with the B's standing directly underneath.

The model consists of a circular template, of radius a, hinged at O, a point on its circumference, to a bar OA, where OA = a. P is the centre of the circle, and E is the middle point of OP. G is a point on the circumference such that OP = OC = a, so that CE is perpendicular to OP.

From E two arms EX and EY radiate, and are no arranged by a linkage that the angles YEC and XEC are equal.

In 1901, when I was drawing up notes on Mental Arithmetic, I looked into many text-books in search of a simple method for dividing by such numbers as 19, 29, 99, 87, etc., but found none. The following method, viz., that of using a multiple of ten as divisor instead of a given divisor, was then discovered by me, and I think it simple enough to be learned and practised by any one.

If it be required to divide A by D, let the quotient at any step be Q, then the product at that step will be DQ, and the remainder, R = A - DQ, where R cannot exceed D nor be less than zero.

The methods explained here are applicable to a large number of problems relating to the symmetric algebraic functions of n letters, and the special results here deduced from them are merely specimens to indicate some of the ways of applying these methods.

In the first section the main principle adopted is that of taking the standard form of a symmetric function to be a sum extending over all the cases of the typical term got by permuting the letters involved in all possible ways, whether they are different or not; and the main result reached is an Inequality Theorem arrived at by expressing the excess of the greater over the less in an explicitly positive form.

1. Let

where D and E are integers, and C a positive integer not a perfect square, k being the number of partial quotients in the non-recurring part of the continued fraction, and c the number in the cycle,

and let