led to think that a brief geometrical explanation of the chief phenomena would be welcomed by students of Physical Science. It is shown in works on waves that the motion of any particle may be defined by the relations where x and y denote the coördinate of the particle when undisturbed by wave motion, and t the time reckoned from some arbitrary epoch. The functions p (x, t) and 4 (y, t) represent peri. odic oscillations about a mean position, which may be best represented by cosines of angles. x = x + (x) cos (nt-vx)) (2) It is shown that these equations* may always be transformed into where y is the depth of the fluid, and λ is the wave length. When the waves are long compared to the depth of the sea, as in the case of the tide-wave. E r'2 1. which is the where d, a', b', are very small quantities. By means of equa tion (4) equation (3) thus gives equation of an ellipse. As b' is very much smaller than a', the Ellipse is obviously very eccentric, or the horizontal motion is very much larger than the vertical motion, as is well known in the case of tides. It is shown in works on the Mathematical Theory of Tides that the first and chief term of the tide-generating potential is "On the General Theory of Tides and on their Secular Effects upon the Figures and Motions of the Heavenly Bodies," by the author; or Airy's "Tides and Waves." Where m is the Moon's mass, pits distance, r the radius of the Earth, & the geocentric angular distance of the disturbed particle from the Moon. If we differentiate this expression relative to any direction, we shall get the forces which disturb the particle relative to the center of the Earth. The whole theory of the tide is based upon the development and extension of this formula. If we differentiate V with respect to ∞ and put o or = 0, we shall have the condi tion that the force is entirely horizontal, and does not tend to elevate the water. phere under the Moon, from = 0, to w = 54.°75, the forces tend to elevate the water; then in a zone from ∞ = 54.°75 to ∞ = 121.°2, the water is under a depressing influence; from ∞ = 121.°2 to 180° the forces again tend to raise the water. The following figures illustrate the radial and tangential forces which act upon the water. The rise of the water is due principally to the action of the horizontal forces, as the vertical forces are of little effect against gravity. The ebb and flow of the tide is chiefly a horizontal oscillation of the water, the fluid either side of the point of highest elevation having run towards that point to raise the level, and having run away from both sides of the lowest depression to produce the drop. Let us now assume the Moon to move in the Celestial Equator and imagine an aqueous canal of uniform depth encircling the equator of the Earth, and let us then inquire in what manner the water is running in the different parts of the canal at any instant when tides are generated in the fluid by the disturbing action of the Moon. Since in case of the semi-diurnal tides there are two approximately equal tides each day, it is evi dent that the canal will be divided into four parts; first, an elevation extending over an arc of 109°.5; second, a depression extending over 70°.5; third, a second elevation extending over 109°.5; and fourth, a second depression extending over an arc of 70°.5. Suppose we draw around the accompanying equatorial section of the earth a series of ellipses representing the oscillations performed by the particles, and indicate in each ellipse the approximate radius vector and phase of the particle at a given instant. To render these ellipses visible to the eye they are necessarily made much rounder than the paths actually described by the particles of our seas. In the figure let h and h' be the places of high water, where the flow is exactly horizontal and in the direction of the propagation of the tide-wave; let f and f'* denote the two points where the water is falling most rapidly, and the motion of the particles is upward; and finally let 1 and I be the two points where the water is lowest, and the motion directly contrary to the direction of the motion of the tide-wave. The arcs between these points are as follows: hf=54.7° lr= 35.3° th' 54.7° * The prime (') is omitted in cut; should appear with f on lower right hand. These arcs are deduced from the tide-generating potential, and represent the parts of the circumference over which the tide-generating forces tend to elevate or depress the water. Assuming Rotation of Earth Tide-Wave solid mustour of the earth ་་་་ that the fluid in the equatorial canal will respond to the forces acting upon it, we have inserted in the figure a series of ellipses with radii vectores showing the phase of the particles in the several regions, and also arrows showing the direction of the revolution of the ellipses, as the earth rotates and the tide-wave advances westward. A slight displacement of each particle in the direction of the advance of the tide-wave is seen to account for the motion of the shape, and in this way the major axis of the figure of the canal appears to revolve relative to the earth. The figure thus appears to explain the progress of the tide-wave in perfect accord with the preceding analytical theory. It will be seen that on each side of r the water is running toward that point over a total arc of 180°; 109.5° of the arc being on one side of r and 70.5° on the other; while on the other side of f the water is receding over similar arcs. At 1 or I', the center of the arc of 70.5°, the motion is exactly horizontal and backwards; at h or h', the centre of the arc of 109.5°, the motion is exactly horizontal and forwards. It may be observed that the tide-wave moves in the direction of the flow of the larger of the two arcs which contribute to raise or lower the water at r or r', for f'. This consideration holds for the propagation of a free wave, such as the tide-wave becomes when once generated. Forced waves are of the same general character, but may be propagated at any velocity consistent with the action of the generating forces. Our actual tides are summations of a series of free and forced waves, superposed by the recurring periodic forces of the Sun and Moon; and as the ocean is of variable depth, the two series of waves are propagated at very unequal rates. If we consider the forced state of the sea following the Moon's transit, by a certain interval, it will correspond approximately to the foregoing figure of the tide wave. NEW FORMS OF TELESCOPES AND OTHER OPTICAL IN STRUMENTS. W. B. MUSSON.* FOR POPULAR ASTRONOMY. There have recently appeared in the newspaper and scientific press communications dealing with a new form of telescope, the fundamental principle of which consists in attaining the achromatism of a large object glass by the interposition of a small concavo-convex lens, silvered on the back. The curvatures of the small correcting negative lens, with its internal reflecting surface being so proportioned as to cause its negative chromatism to correct the positive chromatism of the object glass more completely than has yet been attained, at the same time at a much less cost and with a shorter tube. It has recently been announced that Professor Schupmann of the Technische Hoch Schule at Aix-la-Chapelle in Prussia, on July 30, 1897, applied for, and obtained in the United States (under Letters Patent No. 620978, dated March 14th, 1899,) a patent on such an instrument as is aboved described under the name of "Medial-Fernohr." Among other things, it is claimed that one of the advantages of the telescope is that single crown glass lenses alone may be used. Prof. Schupmann, it is also announced, has published a book on the subject. Under the circumstances, and in defence of the interests of two of its members, Messrs. Z. M. and J. R. Collins, the Astronomical Society of Toronto, thinks it proper to intervene for the purpose of laying before your readers certain facts not hitherto published, and which may tend to place the alleged new invention in a different light. *Secretary Toronto Astronomical Society. |