fessor William Thomson (made in 1858, but not yet published), as containing the demonstration of the general principles of the flow of a liquid past a solid body.

3. Every figure of a solid, past which a liquid is capable of flowing smoothly, generates an endless series of water-lines, which become sharper in their forms as they are more distant from the primitive water-line of the solid. The only exact water-lines whose forms have hitherto been completely investigated, are those generated by the cylinder in two dimensions, and by the sphere in three dimensions. In addition to what is already known of those lines, the author points out that, when a cylinder moves through still water, the orbit of each particle of water is one loop of an elastic curve.

4. The profiles of waves have been used with success in practice as waterlines for ships, first by Mr. Scott Russell (for the explanation of whose system the author refers to the Transactions of the Institution of Naval Architects for 1860–62), and afterwards by others. As to the frictional resistance of vessels having such lines, the author refers to his own papers -one read to the British Association in 1861, and printed in various engineering journals, and another read to the Royal Society in 1862, and printed in the Philosophical Transactions. Viewed as plane water-lines, however, the profiles of waves are not exact, but approximate; for the "solitary wave of translation," investigated experimentally by Mr. Scott Russell (Reports of the British Association, 1844), and mathematically by Mr. Earnshaw (Camb. Trans. 1845), is strictly applicable to a channel of limited dimensions only, and the trochoidal form belongs properly to an endless series of waves, whereas a ship is a solitary body.

5. The author proceeds to investigate and explain the properties of a class of water-lines comprising an endless variety of forms and proportions. In each series of such lines, the primitive water-line is a particular sort of oval, characterized by this property, that the ordinate at any point of the oval is proportional to the angle between two lines drawn from that point to two foci. Ovals of this class differ from ellipses in being considerably fuller at the ends and flatter at the sides.

6. The length of the oval may bear any proportion to its breadth, from equality (when the oval becomes a circle) to infinity.

7. Each oval generates an endless series of water-lines, which become sharper in figure as they are further from the oval*. In each of those derived lines, the excess of the ordinate at a given point above a certain minimum value is proportional to the angle between a pair of lines drawn from that point to the two foci.

8. There is thus an endless series of ovals, each generating an endless series of water-lines; and amongst those figures, a continuous or fair" curve can always be found combining any proportion of length to breadth,

* As a convenient and significant name for these water-lines, the term "Oögenous Neoïds" is proposed (from 'Qoyevs, generated from an egg, or oval).

from equality to infinity, with any degree of fullness or fineness of entrance, from absolute bluffness to a knife-edge.

9. The lines thus obtained present striking likenesses to those at which naval architects have arrived through practical experience; and every successful model in existing vessels can be closely imitated by means of them. 10. Any series of water-lines, including the primitive oval, are easily and quickly constructed with the ruler and compasses.

11. The author shows how to construct two algebraic curves traversing certain important points in the water-lines, which are exactly similar for all water-lines of this class. One is a rectangular hyperbola, having its vertex at the end of the oval. It traverses all the points at which the motion of the particles, in still water, is at right angles to the water-lines. The other is a curve of the fourth order, having two branches, one of which traverses a series of points, at each of which the velocity of gliding of the particles of water along the water-line is less than at any other point on the same water-line; while the other branch traverses a series of points, at each of which the velocity of gliding is greater than at any other point on the same water-line.

12. A certain point in the second branch of that curve divides each series of water-lines into two classes,-those which lie within that point having three points of minimum and two of maximum velocity of gliding, while every water-line which passes through or beyond the same point has only two points of minimum and one of maximum velocity of gliding. Hence the latter class of lines cause less commotion in the water than the former. 13. On the water-line which traverses the point of division itself, the velocity of gliding changes more gradually than on any other water-line having the same proportion of length to breadth. Water-lines possessing this character can be constructed with any proportion of length to breadth, from 3 (which gives an oval) to infinity. The finer of those lines are found to be nearly approximated to by wave-lines, but are less hollow at the bow than wave-lines are.

14. The author shows how horizontal water-lines at the bow, drawn according to this system, may be combined with vertical plane lines of motion for the water at the stern, if desired by the naval architect.

15. In this, as in every system of water-lines, a certain relation (according to a principle first pointed out by Mr. Scott Russell) must be preserved between the form and dimensions of the bow and the maximum speed of the ship, in order that the appreciable resistance may be wholly frictional and proportional to the square of the velocity (as the experimental researches of Mr. J. R. Napier and the author have shown it to be in wellformed ships), and may not be augmented by terms increasing as the fourth and higher powers of the velocity, through the action of vertical disturbances

of the water.

VOL. XIII.

с

III. "On the degree of uncertainty which Local Attraction, if not allowed for, occasions in the Map of a Country, and in the mean figure of the Earth as determined by Geodesy: a method of obtaining the mean figure free from ambiguity, from a comparison of the Anglo-Gallic, Russian, and Indian Arcs: and speculations on the Constitution of the Earth's Crust." By the Venerable J. H. PRATT, Archdeacon of Calcutta. Communicated by Professor STOKES, Sec. R.S. Received Oct. 5, 1863.

(Abstract.)

After referring to a former paper in which he had shown that, in the Great Indian Arc of meridian, deflections of the plumb-line amounting to as much as 20" or 30" would be produced if there were no sources of compensation in variations of density beneath the surface of the earth, and after alluding to a remarkable local deflection which M. Otto Struve had discovered in the neighbourhood of Moscow, the author proceeds to consider, in the first instance, the effect of local attraction in mapping a country according to the method followed by geodesists, in which differences of latitude and longitude are determined by means of the measured lengths of arcs, by substituting these lengths and the observed middle latitudes in the known trigonometrical formulæ, using the mean figure of the earth, although the actual level surface may differ from that belonging to the mean figure in consequence of local attraction. He concludes that no sensible error is thus introduced, either in latitude or longitude, if the arc do not exceed 121° of latitude or 15° of longitude in extent, but that the position of the map thus formed on the terrestrial spheroid will be uncertain to the extent of the deflection due to local attraction at the station used for fixing that position. In the Great Indian Arc this displacement might amount to half a mile if the deflections were as great as those calculated from the attraction of the mountains and the defect of attraction of the ocean, irrespective of subjacent variations of density; but the author shows in the next two sections that some cause of compensation exists which would rarely allow the actual uncertainty to be of any considerable amount, unless the station used for fixing the map were obviously situated in a most disadvantageous position.

The author then proceeds to examine the effect of local attraction on the mean figure of the earth, considering more particularly the eight arcs which have been employed for the purpose in the volume of the British Ordnance Survey. He supposes the reference station of each arc to be affected to an unknown extent by local attraction, and obtains formulæ giving the elements of the mean figure obtained by combining the eight arcs, these formulæ involving eight unknown constants expressing the deviations due to local attraction at each of the selected stations. By substituting reasonable values for the unknown deflections, he shows that local attraction s competent to affect the deduced mean figure to a very sensible extent.

He then institutes a comparison between the results afforded by those three of the eight arcs which are of considerable extent, namely, the AngloGallic, Russian, and Indian Arcs. For each arc in particular he deduces values of the principal semiaxes of the earth, involving an unknown constant expressing the effect of local attraction at the reference station of the arc. In order that the three pairs of semiaxes should agree, there are four equations to be satisfied by means of three disposable quantities (namely, the three unknown attractions). On combining these four equations by the method of least squares, the unknown deflections come out extremely small, and the values of each semiaxis deduced for the three arcs separately come out very nearly equal to one another, and therefore to their mean. These mean values the author ventures to assume are the mean semiaxes of the earth. They are as follows:

1 295'3'

a=20926180, b=20855316 feet, giving €= where a is the equatorial, and 6 the polar semiaxis, and e the ellipticity. The author concludes with certain speculations respecting the constitution of the earth's crust. On adopting the mean figure determined as above explained, the errors of latitude to be attributed to local attraction at each of the fifty-five stations of the eight arcs, which will be found at p. 766 of the Ordnance Survey volume, come out very small. With respect to the Great Indian Arc, it is especially remarkable that the residual deflections are insignificant, while those calculated from the action of the causes visibly at work are considerable. It would seem as if some general cause were at work to increase the density under the ocean, and diminish the density under mountainous tracts of country. The author conceives that, as the earth cooled down from a state of fusion sufficiently to allow a permanent crust to be formed, those regions where the crust contracted became basins into which the waters ran, while regions where expansion accompanied solidification became elevated without any consequent increase in the total quantity of matter in a vertical column extending from the surface down to a given surface of equal pressure in the yet viscous mass below. The author considers that the deviations of latitude at the other principal stations of the measured arcs, if not positively confirmatory of, are at least not opposed to this view.

IV. "On the Meteorological Results shown by the Self-registering Instruments at Greenwich during the extraordinary Storm of October 30, 1863." By JAMES GLAISHER, F.R.S., F.R.A.S., &c. Received November 23, 1863.

In the year 1841 Osler's anemometer was erected at the Royal Observatory, Greenwich, and from that time, up to the year 1860, the greatest pressure on the square foot recorded was 25 lbs. In February 1860 one of 28 lbs. was registered, which was the greatest up to October 30 of the present year; on that day a pressure of no less than 294 lbs. took place

during a heavy squall of wind and rain, which passed over the observatory at 3h. 30m. P.M. At this time, moreover, the readings of the several other self-registering meteorological instruments at the Royal Observatory, Greenwich, exhibited very large changes, and of so remarkable a character, that the Astronomer Royal expressed a wish that I should bring them under the notice of the Royal Society. The following are extracts from the several registers of the day mentioned :

At 6h. A.M., on October 30, the barometer read 29-32 in., and it commenced falling slowly after this time, reaching 29.30 in. at 8h. A.M. The decrease then became more decided, and a steady fall was experienced ; 29.10 in. was reached by 04h. P.M., and 28.96 in. by 2h. P.M.; from 2h. P.M. to 31⁄2h. P.M. the decline was very rapid; and the minimum reading, 28.80 in., was reached at the latter time.

After 3h. 30m. P.M. the barometer turned to increase rapidly; at 3h. 39m. P.M. it read 28.85 in.; at 4h. P.M., 28.92 in.; 4h. 20m. P.M., 29.00 in.; at 5h. P.M., 29'07 in.; and afterwards a gradual increase took place to 29-30 in. by 11h. P.M.

At 8h. A.M., with the first indications of decided barometric fall, the wind commenced to blow strongly from S.W.; at 8h. 20m. A.M. it had reached a force of 1 lb. on the square foot; shortly after, 2 lbs., and at 8h. 30m. A.M. 3 lbs. A force of 1 lb. to 3 lbs generally prevailed, till 9h. 25m. A.M.; at 9h. 30m. A.M. a gust of short duration was experienced of 15 lbs., which produced a decline of temperature of 2°. From 9h. 35m. A.M. to 9h. 50m. A.M. the pressure of the wind varied between 3 lbs. and 5lbs. ; from 14 lb. to 3 lbs. from 9h. 59m. A.M. to 0h. 45m. P.M.; there was no pressure for two or three minutes about 0h. 50m. P.M.; the wind then again commenced blowing strongly, reaching 4 lbs. at Oh. 55m. P.M., and from 3 lbs. to 5 lbs. from 1h. P.M. to 1h. 15m. P.M., the pressure was generally 2 lbs. to 4 lbs. from 1h. 15m. P.M. to 2h. P.M.; from 2h. 0m. P.M. to 2h. 45m. P.M. it varied between 4 lb. and 2 lbs.; the wind again commenced blowing strongly, reached 31⁄2 lbs. at 2h. 50m. P.M., 4 lbs. at 3h. P.M., 5 lbs. at 3h. 10m. P.M., 7 lbs. at 3h. 16m. P.M., 12 lbs. at 3h. 20m. P.M., 13 lbs. at 3h. 23m. P.M., 11 lbs. at 3h. 26m. P.M., 17 lbs. at 3h. 29m. P.M., and 29 lbs. at 3h. 30m. P.M.; then declined suddenly, pressing with forces varying between 6 lbs. and 9 lbs. from 3h. 35m. P.M. to 3h. 45m. P.M., and 4 lbs. to 6 lbs. from 3h. 45m. P.M. to 4h. P.M.; another gust at 4h. 10m. P.M. reached 8 lbs.; again declined to 4 lbs. at 4h. 15m. P.M.; after this time, till 5h. P.M., the pressure varied between 2 lbs. and 4 lbs., between 2 lbs. and 3 lbs. from 5h. P.M. to 6h. P.M., from 2 lb. to 2 lbs. (with occasional lulls) from 6h. P.M. to 7h. P.M., from 2 lbs. to 4 lbs. from 7h. P.M. to 9h. P.M.; scarcely any pressure was recorded between 9h. P.M. and 10h. P.M., and from 10h. P.M. to 11h. P.M. the amount varied between 14 lb. to 3 lbs.

At the time of the great gust, viz. 3h. 30m. P.M., the barometer reached its minimum, 28.80 in.; the temperature declined rapidly (from