The following formula

is given by Laplace (Theorie Analytique des Probabilités, 2 ed. Paris, 1814, p. 195) as an approximate value of Δn0i/ni, when n and i are very large numbers, and is applied immediately afterwards to the case where i is of the order n log n.

In designing a telescope for a particular class of observations, it was found desirable to have a field-bar in the focus of the object-glass, and, at the same time, to have an image of that object-glass exterior to the eye-lens. These desiderata cannot both be got by the achromatic arrangement of two lenses made of one material; they are combined in the ordinary four-lens eye-piece.

At night the cobwebs of the telescope are invisible for want of light; we have to illuminate either the field or the wires, so as to make them visible by contrast. At noon the astronomer meets with the same kind of difficulty from an opposite cause; the intense sun's light has to be obscured by a dark glass, which, at the same time, completely obliterates the spider-lines; these are only seen on the sun's face. In consequenoe, the advent of the sun's edge to a wire cannot be observed ; the line must be fairly on the sun's face before we can see it, and thus the noted instant is necessarily too late,—too late by a quantity depending on the power of the telescope and on the skill of the observer. Hence the estimate of the sun's apparent diameter from observations of the meridian passage may be expected to err slightly in defect, while the thence-deduced right ascension must be too great.

The present paper contains a description of five apparently new species of Lumbricidæ from Australia and New Zealand, one of these species being perhaps the type of a new genus, which I have named Neodrilus; the remaining species are Acanthodrilus neglectus, from New Zealand, Perichæta newcombei, Urochæta, sp. ?, from Australia, and P. upoluensis, from one of the Pacific islands. I have endeavoured to make these descriptions as full as the material, in many cases in an excellent state of preservation, has enabled me to do. I have also incorporated into this paper some few notes on Perichæta antaretica, Baird, a species which has not yet been sufficiently discriminated.

The observations recorded in this paper have reference chiefly to the coast-sections at St Abb's Head and Coldingham Shore. The district was geologically surveyed some twenty-five years ago by my brother, Dr A. Geikie, and subsequently described by him in the Memoirs of the Geological Survey. Since the publication of that memoir, no further examination of the ground in question appears to have been made. During the past summer I visited the neighbourhood, principally for the purpose of studying the igneous rocks which are so well exposed in sea-coast sections. At the date of the Government Survey of Eastern Berwickshire the aid of the microscope had not yet been invoked by field-geologists for the purpose of determining rock-species, and I was therefore curious to compare the igneous rocks of that region with those of similar age which I had studied elsewhere in Scotland, and more especially with the bedded and intrusive porphyrites and tuffs of the Cheviot Hills and the Sidlaws.

The summations referred to in the title depend on two theorems—one already known, the other not hitherto published. The first of these is to the effect that the quotient of any simple Alternant by the difference-product of the variables is expressible as a determinant whose elements are sums of homogeneous products of the variables.