and these give the motion of translation when the point (x', y', z') is assumed for origin From these equations we have

Hence, if the velocity of translation be resolved into two, respectively perpendicular to and along the axis of rotation, the latter remains invariable in magnitude whatever origin be adopted; and hence also the velocity of translation will be the least possible when the origin is so assumed that the former vanishes, or, in other words, when the velocity of translation is in the direction of the axis of Rotation.

If we seek an origin which shall make the motion of translation vanish, or which shall make the whole motion reducible to a single rotation, we must have for the determination of this origin (x', y', '),

or

u'

o, v'

0,w' = 0,

Those equations are inconsistent unless a certain condition hold,

which is

+ v w +

y

(2)

0 = u w x When this condition is satisfied (provided w, wy, W% o do not all vanish), the equations (1) are equivalent to only two independent equations and represent a straight line, every point of which is an origin such as required.

In the particular case where the motion consists of rotations round parallel axes, taking o as the type of one of these about an axis through the point (x, y, z), and l, m, n for the direction-cosines of the common direction of their axes, we have

n Σ(w).

Also the linear velocity along the axis of a generated in the origin of co-ordinates by one of these rotations w being n w y

The condition (2) is in this case satisfied, and (provided (w) do

not vanish) the equations (1) assume the form

Hence the axis of the single resultant rotation passes through the

point whose co-ordinates are

(wx) (wy) (wz) Σ(w) Σ(m)' (w)

-; these are independent of l, m, n, and this point therefore remains the same so long as the magnitudes of the rotations, and the points through which their axes are drawn are unaltered, whatever be the common direction of these axes; by analogy this point might be called the Centre of Parallel Rotations.

ON A SMALL CAPILLARY WAVE NOT HITHERTO

DESCRIBED.

BY JOHN LANGTON, M. A.,

VICE-CHANCELLOR OF THE UNIVERSITY OF TORONTO.

Read before the Canadian Institute, 17th January, 1857.

It is well known that the shape and velocity of waves, and the different circumstances under which they are propagated, have attracted considerable attention amongst men of science, not only from the importance of the subject as connected with the theory of tides, but also from its practical bearing in relation to the resistance of fluids, and the best form for vessels which are destined to move in them. An elaborate report upon waves was prepared by J. Scott Russell, for the British Association, in 1844, the experiments detailed in which have been the origin of some of the greatest improvements of the present day in ship-building, and have inseparably connected the wave line with the name of Russell. Although this report is principally devoted to the solitary wave of translation, which gave rise to the investigation, it treats at less length of other varieties, and may, I believe, be said to embody all that is known upon the subject from observation. There is nothing, however, amongst the waves there enumerated in any way resembling that which I propose bringing under the notice of the Institute to-night, nor have I elsewhere seen any account of its having been previously observed. Amongst all the different kinds of waves, varying as they do in dimensions from the great tidal wave, which, with an elevation of only a few feet, rests upon a base of hundreds of miles in extent, to the ripple which is raised by a summer's breeze, the wave which I am about to introduce to you is, perhaps, the smallest. It

is, indeed, so minute that it might easily have escaped notice, for under ordinary circumstances it cannot much exceed of an inch in height, and hardly reaches an inch in amplitude. Such an exceedingly minute object may seem scarcely of sufficient interest to form the subject of a paper to be read before the Institute; and being apparently only a disturbance of the capillary film on the surface, which is subject to very different laws and forces from those which govern the motions of the whole mass of a fluid, it is doubtless of much inferior importance to the waves which Russell experimented upon. But as no fact is so trifling that it may not assist in establishing correct views of the operations of nature, and as the Institute has invited communications from its members, giving an account of original observations upon all phenomena, I venture to call attention to some curious particulars which I have noticed respecting this capillary wave, which differs from all others previously described in being a solitary one.

The wave in question may be observed in three different situations. Wherever a large body of water, with a strong current, meets comparatively still and deep water below, it may always be seen, as a sharply defined line, like a hair upon the surface, winding about amongst the numerous eddies which are formed in such situations, ever varying in its outline, and carried along apparently with the general course of the stream, whilst upon the whole it maintains nearly the same position. It is also generally to be found where there exists any impediment to a current, as a dead tree projecting out into a river, or a boom thrown across the stream. In this case the wave may be observed at a distance of from one to three feet above the obstacle, the distance varying with the force of the current. The third case is the reverse of the former one, where a body, propelled through still water, pushes this small wave before it. It occurs much more rarely under these circumstances, and may more easily elude observation, and since my attention has been attracted to it I have often failed to produce it; but it was in this form that I first got any insight into its nature.

Paddling in a canoe in a sheltered bay, with just sufficient air stirring overhead, without raising a ripple on the water, to cause the canoe, when abandoned to itself, to drift broadside on at the rate of perhaps half a mile an hour, I perceived the wave in advance of the canoe, at a distance of about three feat. If the wind died away, the wave was maintained at a greater distance, and upon one occasion I could distinctly trace it so far as from between eight and nine feet

VOL. II.-G

Uor M

from the canoe, beyond which it became fragmentary, and disappeared. If the speed increased suddenly, the wave disappeared, and the slightest ripple on the surface obliterated it at once. But if the wind freshened very gradually, the wave approached nearer and nearer, becoming more strongly defined and more elevated above the surface, and it could still be seen under the lee of the canoe, and at a distance of eight or nine inches from it, after the breeze had increased so as to make a strong ripple on the water outside. If the speed grew still greater, it became first obscured, and finally destroyed, by waves of an entirely different character,-viz., the ripple caused by the canoe itself,-which it is remarkable did not make their appearance in contact with the canoe, but first broke out on the farther side of the capillary wave.

But my attention was principally attracted to the effect which was produced upon the little particles floating near the surface. The opportunity was a good one for observation, for, being in the vicinity of a marsh, the water was very impure, and a bright sunshine enabled me to see the motes at a considerable distance. Light bodies resting on the water without being wetted, as a feather or thistle down, seemed hardly at all affected, except by a slight motion as the wave passed under them; and larger particles, which reached to a depth of perhaps an eighth of an inch, passed it without any disturbance. But smaller ones, floating close to the surface, were violently agitated as the wave reached them, and though they passed a little beyond it, their apparent motion towards the canoe was retarded and finally stopped, at distances from the wave depending upon their respective sizes, the larger ones penetrating the farthest. The interval between the wave and the canoe became thus soon filled with small objects, very regularly sorted according to their sizes, the larger, however, being proportionally much closer together than the smaller ones. If the wind now freshened the wave approached nearer to the canoe, and all the particles were driven in with it, but the smaller ones were much more affected than the larger. If the wind again slackened the wave receded, leaving the particles where they were, and fresh ones were collected in the vacant space, so that, after a few alterations of speed, the regular assortment according to size was soon interfered with, and a miscellaneous scum was pushed on before the canoe, comprising floating matter of all sizes, up to an eighth of an inch or more in depth; for it must be observed, that although objects of that size passed under the wave without disturbance, and penetrated

SARY

a long way beyond it without check, their motion was at last arrested before reaching the canoe.

From these facts I was induced to conclude that a body propelled through the water at a low velocity, and with an even regular motion, pushes before it a wedge-shaped film of water, the under surface of which is not a straight line, but a curve of rapidly increasing curvature; that, at very low velocities, this film remains unbroken to a distance of several feet; and that, with increased speed, the distance to which it extends is diminished, whilst its greatest depth remains nearly uniform. There are two things, however, which this supposition will not account for. It will not account for the wave itself: for the film which I have imagined does not appear to extend so far, and under no circumstances did any of the particles, even the smallest, become stationary, till they had passed the wave by about two inches, that interval being always perfectly free from the scum collected. The other circumstance which is left unexplained is, why the feathers and thistle down resting on the water were not also arrested when they came to the film. I do not attempt to account for the difficulty, I only record the facts as I observed them.

I endeavoured to arrive at some conclusion as to the form and size of the wave, but without much success. Its exceeding minuteness, the inconvenient position of the observer in the canoe at a considerable distance from it, and that distance constantly changing with the varying force of the wind, made any accurate measurements almost impossible. I therefore had recourse to the second form, in which I have mentioned that it occurs, where, from the similarity of the circumstances, one would expect to meet with the same facts, and which in many respects afforded greater facilities for observation.

When the water was high in the river in which I made my observations, a great deal of foam came down from the falls above, and at every projecting tree there was a dense collection of froth, with a clear space intervening between it and the wave. Upon clearing away this froth I could observe its gradual re-formation. It was very curious to watch a small patch of foam sailing down with the current. When it approached very close, and in passing the wave, its velocity seemed momentarily increased, but it was then suddenly arrested, whilst there would shoot out from underneath it bits of sawdust, and other matter that had become entangled in it, which would arrange themselves according to their draught of water in the vacant space between the wave and the log, the foam itself remaining