I have to explain at the outset that the title of my paper has been chosen for the sake of brevity. Both the words “expiatory” and “substitutionary” are liable to be misunderstood and abused in argument. I shall, therefore, avoid the word “expiate” as much as possible; but as I cannot do without a frequent use of the words “substitution” and “substitutionary,” I merely note at present that there are various modes of substitution and various purposes served by it, and that it is very important in an argument to state accurately both the mode and the purpose.

The object of my paper is to place the two prevalent theories of the origin of sacrifice in juxtaposition with the facts which we have in regard to Greek sacrifices, in order that we may see what light the facts may throw on the theories. For this purpose I first state the two theories.

In Crelle's Journal, vol. x., Stern devotes fifteen pages (pp. 245-259) of his Theorie der Kettenbrüche to the examination and elucidation of the following problems:—

as continued fractions of the form

where a1, a2, a3, … are independent of x.

§ 1. The general problem in elasticity, as usually presented for solution, supposes the elastic substance to pass initially from a state without strain. But important cases exist where it must be conceived to start from a state already under considerable stress. When this is the case, the constitution of the solid undergoes great change, as is shown by the fact that strained glass loses its isotropic property, and becomes doubly refractive. This subject was long ago attacked by Cauchy, who, by means of the theory of molecular actions, deduced the existence, in the expressions for the stresses due to the secondary strains, of terms proportional to the initial or primary stresses. The problem has been since discussed by MM. De St Venant and Boussinesq, who have applied to it Green's expression for the energy stored up during the strain. But the question, in their hands, still retains traces of Cauchy's hypothetical element, inasmuch as their expression for the potential energy was deduced by means of the molecular theory. M. De St Venant even considers it a strong argument for the truth of the latter, that it is indispensable in the discussion of this problem. These authors have also failed to see in what way the remaining part of the potential depends on the original strain.

In the series of chapters of which the present is the first, I purpose to submit the results of an analytical examination of all the minerals of Scotland whose composition appeared doubtful; of such as had not previously been examined; or of such as appeared in any way to be of special geologic interest.

A few years ago, a memoir on high-water marks on the banks of the Rivers Earn and Teith, in Perthshire, by the Rev. Thomas Brown, was read in our Society, and published in our Transactions.

The only other Scotch geologist, so far as I know, who has alluded to the existence of river terraces, much above the level of existing floods, is the late Dr Robert Chambers. In his work, entitled “Ancient Sea Margins,” Dr Chambers specifies many Scotch rivers, in the valleys occupied by which, he had seen terraces, at considerable heights above the rivers and above the sea.

The presentation to the Royal Society of Edinburgh of results relating to the decennial period, derived from observations of magnetic declination made during nearly a quarter of a century at Trevandrum, has seemed to me a favourable occasion for a determination of the mean duration of this period. Upon the explanation of the decennial variations depends the solution of several important problems in solar and terrestrial physics, and the first step towards this result is to ascertain the true mean duration of the period. Two markedly different results have been obtained, each of which has been accepted by men of the highest scientific reputation.

I. The Parallel Roads of Lochaber have presented to geologists a problem, which is still unsolved. Dr Macculloch, about sixty years ago, when President of the Geological Society of London, first called attention to these peculiar markings on the Lochaber Hills, by an elaborate Memoir afterwards published in that Society's Transactions. He was followed by Sir Thomas Dick Lauder, who in the year 1824, read a paper in our own Society, illustrated by excellent sketches. His paper is in our Transactions. The next author who attempted a solution was the present Mr Charles Darwin. He maintained that these Roads were sea-beaches, formed, when this part of Europe was rising from beneath the Ocean. He was followed by Professor Agassiz, Dr Buckland, Charles Babbage, Sir John Lubbock, Robert Chambers, Professor Rogers, Sir George M'Kenzie, Mr Jamieson of Ellon, Professor Nicol, Mr Bryce of Glasgow, Mr Watson, and Mr Jolly of Inverness. Sir Charles Lyell, though he wrote no special memoir, treated the subject pretty fully in his works, giving an opinion in support of the views of Agassiz.

In referring to the tendency to bilateral ramification on the branches and even on the main stem in Cupressineæ, such as Cupressus, Thuja, and Libocedrus, Professor Sachs, in his Text-book (Bennett's translation pp. 444–5), thus expresses himself: “Branch systems of three or four orders of shoots are developed in one plane in such a manner that a system of this kind assumes a definite contour, and somewhat the appearance of a pinnate leaf. In Taxodium the foliage-leaves are formed in two rows on slender branches a few inches in length; in T. distichum these fall off in autumn, together with their leaves, thus presenting a still greater resemblance to pinnate leaves.”