^{page 544 note *} Professor Tait has urged me to make known a reminiscence of my youth that at the time here referred to there were in Edinburgh, and in this Society, no fewer than three mathematical amateurs, who, though they never made themselves publicly felt as such, in some measure saved this corner of the land from the censure dealt in the text. These were Sir William Miller, Baronet, of Glenlee, better known as Lord Glenlee of the Scottish bench; William Archibald Cadell, of the family of Cadell of Grange, who finished his earthly career but a few years ago; and my own father, Professor of Latin in our University. Lord Glenlee, a man of very retiring habits and disposition, was usually called the first amateur mathematician in Scotland. Mr Cadell, also a man of great reserve and shyness, nevertheless, in order to carry out his admiration of the modern continental mathematics, contrived to obtain, during the very hottest of our struggles with France, from that generally unyielding potentate, the First Napoleon, permission, through the influence of one of the great mathematicians of Paris, to repair to the French capital, to dwell there for seven years, and to return unhindered to Scotland, at a period when no other Briton was known to have put his foot on French soil without being made a detenu. My father, during the last ten years of his life, which ended in 1820, betook himself, as his idea of relaxation from routine professional life, to the differential calculus, and to Newton, Bernoulli, Euler, Lagrange, Laplace, Lacroix, &c., whose works were always at hand when not in his hands. As he made a vigorous attempt to indoctrinate me at a very early age in his favourite pursuits, I know well what these were, and what he knew of the kindred spirits Glenlee and Cadell.

^{page 549 note *} This method, devised for the occasion, I have since found to be a mere variety, but more convenient, of one proposed some years ago by Dr Letheby, and adopted by the late Professor Miller.

^{page 555 note *} This is a good method of ascertaining the relative colour of waters if it be employed with due precautions. The trial should be made in sunshine— when the sheet of water is quite calm—between 9 A.M. and 3 P.M., SO that the sun's rays may not fall too obliquely on the water, and with the back to the sun, and, best of all, on the shady side of a boat. If all these conditions be reversed, vision will penetrate scarcely half so deep as when they are all observed. In my recent trials I have not found a white object visible at a greater depth than 21 feet, viz., on Loch Lomond on the 6th May. But, from observations made many years ago, I am satisfied that, after long dry weather, some river waters will allow such an object as a white porcelain basin to be seen at a much greater depth, with due attention to the conditions now mentioned. Having a recollection of seeing it stated long ago, that the water of the Lake of Geneva was so clear, that objects fcould be distinguished in it at a very great depth, I applied to Dr Coindet of Geneva for precise information, for which he referred me to Professor Forel of Lausanne. To Professor Forel's kindness I am indebted for the following interesting facts:—In the spring of 1869, using a white-painted sheet of iron, 15 inches by 12, he found that the utmost depth at which it could be seen was 13 metres, or 44 feet. The transparency is much affected by locality, and very much too by season. In winter and spring it is greatest, in summer and autumn least. In the Bay of Morges, objects may be seen distinctly at the bottom in winter at a depth from 13½ to 20 feet, while in summer they are barely visible through 7 feet. This difference is greatest near the shore, at the bottom of bays, and near villages or towns. It is least around promontories, far from land, and at a distance from human habitations. In autumn the change from obscurity to transparency usually takes place early in October, and is completed in three days; in summer, the reverse change takes place about the beginning of May, and is more gradual. By filtering a large quantity of turbid water, he found the obscuring cause to be a collection of amorphous dust, living and dead diatoms, vegetable debris, a few living infusoria and crustaceans, and debris of insect larvæ and microscopic crustacea. They naturally collect slowly in the summer; but the first cold of approaching winter sends them quickly down with the water as it cools.

In the case of Loch Lomond, these inquiries of Professor Forel would lead one to expect little influence from organic or inorganic dust in obscuring water where it is so deep as at the places chosen for my observations. Accordingly, the surface water was remarkably free from turbidity, or deposit on standing at rest. But the yellowish colour, faint though it be, constitutes a no less powerful obstruction to the penetration of light. The depth of colour, and consequently the transparency, vary at different periods, not so much with the seasons as with the times of floods. In advanced summer and in autumn, the floods increase the colour decidedly, and lessen for a time transparency. But my single observation on 6th May, when I found the transparency greatest of all a few days after heavy north-east rain, raises a question whether floods have the same effect in spring or the end of winter. A probable reason for the contrary may be, that the soluble matters of the peat-fields and stream-courses, developed by heat, growth, and atmospheric action in summer and autumn, are much exhausted by the frequent winter floods before the arrival of the floods of spring.

^{page 557 note *} 26,250 grains left 0·83 at 300° F., and 0·62 after incineration.

^{page 558 note *} 13,125 grains left 0·82 grains at 300, and 0·55 after incineration.

^{page 558 note *} The cistern which brought up the water was new, made of copper, and urnished, for valves, with spherical copper balls resting on hemispherical beds, and it was never used except for these experiments. The cistern was emptied at once into stoppered bottles on being drawn into the boat, and was carefully dried in a current of air with the valves open.

^{page 567 note *} It has been said that stones covered with green confervæ and other diatoms do occur in Loch-Lomond. They do in bays and other shallows; but the covering is very thin; and the line of such stones is narrow. Where deep water is near there are none at the edge, and where they do occur the dry stones close to the edge appear quite clean.

^{page 572 note *} In the course of his calculations Mr Buchan arrived at the interesting fact that the average mean temperature of the air during the six cold months of these years, at the level of the lake's surface, was 41°·7 from November 18 to April 10, cr very nearly that of the deep substratum.—see subsequently, for his observations, the later Proceedings of the Society.

^{page 574 note *} While the preceding statements were passing through the press, my attention was called to similar observations in Sir John Leslie's article on Climate in the “Encyclopaedia Britannica,” by Saussure on the Lakes of Geneva, Thun, and Lucerne, and by the late eminent engineer, Mr James Jardine, on Loch Lomond and Loch Katrine in 1814. Their observations are not entirely concordant with those given above. I contemplate further observations which may reconcile them.