= order, illuminated by Fresnel's great lamp, with a wick 3-6 inches in diameter, then, if the internal radius of the mirror be assumed at 24 inches, and if 1.55 for the extreme red rays, we shall find, as formerly, the limiting value of to be 8° 11′ 40′′-8°2 nearly; from which it follows, since = 180° =22, that there must be 11 zones and a central conoid. Hence 7° 49′ 34′′. From this value of, putting d = 24, and f= 1.8, it will be found from the formulæ (1) = r = 70-624; a 24-260; b = 49.940. If r, a, and b, be calculated by means of the osculating circle to the parabola in the manner already explained, it will be found that = 51.278. 35 r = 72·517; a = 25.639; b From this it appears that the values of r, calculated by the two formulæ, differ by nearly 1.89 inch, or by about 3th of the whole length of the radius. The difference of the ordinates of the circles at the point A will be found in the same manner as before, to be 0027 inch, from which the perpendicular distance of the arcs is found to be 0018 inch, a quantity quite within the limits of error in constructing such apparatus. Finally, the error in inclination of the osculating circle to the parabola at the point A is 5′ 46′′, from which the lateral aberration of the rays from the point F will be about 16 inches, a quantity which may be safely neglected with a flame 3.6 inches in diameter. It seems, therefore, from these examples that the approximate formule will give sufficiently accurate results in the cases most likely to occur in practice. It may have already occurred to the reader as a remarkable peculiarity of the totally reflecting mirror, that since the at their first incidence and final emergence are perrays pendicular to the surface of the glass, they suffer no deviation by refraction; and consequently the curvature of the different surfaces is totally independent of the refractive power of the glass. It is otherwise, however, with the determination of the greatest admissible breadth of the zones. Here the index of refraction of the glass enters as an element into the calculation; and the higher the refraction the greater may be the breadth of a zone capable of reflecting all the incident light. Flint glass, from its high refractive power, is therefore more suitable than plate glass for the construction of the totally reflecting mirror; for, owing to their greater breadth, a smaller number of zones of the former material will be required to complete a hemisphere, and the expense of construction will accordingly be lessened. Since this paper was read, it has occurred to me that the totally reflecting mirror will probably have certain properties rendering it preferable to a metallic reflector, which it may be proper to mention here. In a recent application of the holophotal principle to an instrument of very great size, so much heat was reflected from the metallic hemisphere as to cause the ebullition of the oil; an inconvenience which, however, has been obviated mechanically. Now, it has been found that, at moderate angles of incidence, glass reflects much less heat than metal. If this be true also at an incidence of 90°, the totally reflecting mirror, by its superficial reflexion, will return much less heat than a metallic hemisphere. Then, of the heat-rays which enter the glass, besides those which are absorbed, it is probable that a considerable portion, owing to their low refrangibility, will escape total reflexion, and emerge harmlessly at the back of the mirror. The investigation of the different formula already given, is that by which they were originally obtained. The following are the principal steps of geometrical demonstrations, which are added for those who may prefer them. 1. In fig. 2 join AB, and let AD, FB intersect each other in G. Since the ray FA is reflected at A in the direction AC perpendicular to FB, the normal AD bisects the angle FAC. Therefore putting AF=d, AD=r, FE=a, DE=-b, and AFC= CAD DAF (90°- AFC), or DAF 45°— = = GBD+GDB GAF + AFG; Then from which ADB: = = 4' and ABF ABD-FBD=45° FE-EC-FC=AD sin ADE-AF cos AFC, Also and a=r sin (45° d cos and brsin 45°. 4 2. To find the osculating circle at the point B, on the supposition that AB is a parabolic arc, whose focus is F, and BF its principal semi-parameter. Produce BF towards F, until the whole line produced is equal to four times BF; and on it describe a square. About the square describe a circle, which will then be the osculating circle at the point B (see Wallace's Conic Sections, Prop. 1. on Curvature); and putting FB=2m, we shall evidently have the radius of the circle 4 m√2. BE=ED=4m; therefore FE=2 m. Hence, as before, r=4m √2; a=2m; b=-4m. VOL. IV. с On the Improvements on the River Clyde during the past Hundred Years. (Part I.) By WILLIAM CAMPBELL, Esq., C.E.* With a Plate. The progress of improvement on the River Clyde, during the past hundred years, has been from a state of nature, spreading over a wide bed full of shoals, to a highly improved condition; forming a valuable inland navigation and one of the greatest works of the kind. What makes this the more remarkable is, that the extensive forest of masts at the Broomielaw Harbour of Glasgow has accumulated before our own eyes; so rapid has been the progress of improvement since the commencement of the present century. Just one hundred years ago the magistrates of Glasgow had their first report on the levels and depths of the river, down to Dunglas Quay. Since then, about fifty reports have from time to time been had from the principal engineers and others. These reports point out proposed works and embrace the navigation down to Port-Glasgow; in some the remarks extending to the banks at the deep sea. I have to observe that it is chiefly from these, as also from historical and statistical information, joined to my own knowledge of the works, that I have been enabled to attempt to lay before you a general description of what has been done on this important national undertaking; and I have endeavoured to indicate the whole on the maps and plans now before you. On Dumbuck Ford, twelve miles down from Glasgow, there was but 2 feet depth at low water summer level, and the same at Newshot Isle and other points up to a sand at the lower end of the harbour, on which there was but 15 to 18 inches of water. The spring-tides, which in the estuary rose 10 feet, di *Read before the Society, 10th February 1851. minished to 6 feet at Newshot Isle, and to 1 foot 9 inches at the harbour, and here the rise of neap tides died away, so as to be only perceptible at the Broomielaw Quay. Before entering on the details, allow me, Sir, to draw your attention to a general view of the river. The Clyde rises in the south of Lanarkshire, in the same hills above Moffat as the Annan and the Tweed, each taking a different direction. The catchment basin of the Clyde defines the county of Lanark, to which the tributary streams give the appearance of a leaf of which the Clyde is the stem, draining 972 square miles. From the water-shed on Queensberry-hill, the Clyde follows a north-easterly course of seventy miles to Dunbarton Rock, where it receives the Leven and joins the Frith of Clyde, which extends about thirty miles to the little Cumbrae Island, on which is the first lighthouse maintained by the Clyde Ports.* The sources of Clyde are about 2500 feet above the sea ; and near Bothwell, about eight miles above Glasgow, it has fallen to about the level of the sea. About half way between these points where it winds round the base of Tinto, it has fallen to about 700 feet above the sea, and the Falls of Clyde at Lanark descend 180 feet in three miles. At Bothwell the river spreads over a wide rocky bed, and the banks are often flooded up to Dalserf, which is sixteen miles above Glasgow. And finally, there is now only a fall of 1 foot on the surface of the tide below Glasgow, the bottom of the channel being quite level to Port-Glasgow, a distance of eighteen and a quarter miles. The level of the tide in the Frith of Clyde is higher than in the Atlantic Ocean, and up at Bowling Bay it is higher than on the Frith of Forth. The neap tides rise to the level of the weir at Glasgow harbour, and ordinary tides reach Dalmarnock bridge, three miles above. If the weir were removed, and Dalmarnock ford deepened, &c., it seems possible to get the tide to ebb and flow ten or * This lighthouse, erected in 1757, was one of the earliest in Scotland. The light is fixed, and 106 feet above the sea. |