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ordinates of any point in the line PQ will be of different lengths from those of any other point in the same line. But as the line is made according to a regular law, some relation must exist between the co-ordinates, which is the
same for every point in the same line (5,) M O M N though the co-ordinates themselves
And this can only be the case when the variation of one quantity depends upon that of another. An instance of this is found in the common case of two buckets in a well. These may be made to descend together; or, one may go down as the other comes up; or, some other method may be adopted. But whatever the law of their motion be, it is the same for every position of the bucket, though the lengths of the rope vary. Consequently, if the law be known, and the position of one bucket be ascertained, the position of the other may be at once calculated. Or, which is the same thing, if the law is known, the length of one rope may "be expressed in terms of the length of the other." In a similar manner, in the above case the length of one co-ordinate may be expressed in terms of the other, and if these two quantities be represented by the two variables, x and y respectively, this relation assumes the form of an algebraic equation.
For example, the equation y = a represents a straight line such that OM = PM in every position of the point P (6.) And such a mode of expression applies of course not merely to a straight line, but to every line which is formed according to a known law, the law indeed being itself nothing else than this very relation between the co-ordinates, which remains invariable though the co-ordinates themselves vary. Such an equation, i.e., one which expresses the relation between the co-ordinates of every point in a
given curve is called the "Equation to the curve" which it represents. It varies of course in form according to the kind of curve represented by it. Thus an ellipse will have one form of equation, a circle another, a straight line yet another (7, 8;) and so on so that the kind of curve intended to be represented is at once known by the form of the equation. By this method, commonly called that of Rectangular Co-ordinates, the processes of Algebra become applicable to Geometry, and the properties of each particular curve become easily deducible from the equation which represents the relation between the co-ordinates of every point in it. This subject may be read, first analytically in Todhunter's "Conic Sections," then geometrically in Goodwin's "Course of Mathematics," and afterwards analytically a second time in Professor Salmon's "Treatise," a work of great elegance but rather difficult to a beginner.
VARIETIES OF CURVILINEAR MOTION.
THE invention of new laws of motion has been a favourite employment amongst geometers even of the highest eminence both in ancient and modern times. Thus to Archimedes is due the Spiral,
the Cissoid to Diocles a (who invented it for a geometrical purpose,) the Conchoid to Nicomedes, and a most curious curve called the Witch to Agnesi, a lady of surpassing beauty and intellect who filled the chair of mathematics at the University of Bologna in the year A.D. 1750. These curves it may be sufficient merely to name, as to trace them would be quite beyond the scope of the present treatise, and even to enunciate their law of formation would require further knowledge than we at present possess. But there are two which we may notice as being of
common occurrence, viz., the Cate
nary and the Cycloid. The Catenary (1) is the curve in which a perfect flexible string will hang (in vacuo) under the action of gravity.
The Cycloid (2) was first noticed by Descartes, and an account of it pub
lished by Mersenne A.D. 1615 c. It is, in fact, the curve described by a nail in the rim of a carriage-wheel, as it revolves on a horizontal plane. It has the curious property, that if a body fall through any arc of the cycloid reversed, it will fall in the same time, whatever the length of the arc. Consequently, if a pendulum vibrate in a Cycloid, all its vibrations will be performed in equal times, however great or small they may be. This fact is familiar to us all.
This latter curve has suggested to me another of a similar nature, which, so far as I can learn, has not hitherto attracted the notice
(a)-Gregory's Mathematics, p. 176. For these curves and their equations see Price, vol. I.
(b)-Nouvelle Biographie Générale, tom. 1. p. 398.
(c)-Gregory's Mathematics, p. 176.
(d)-Calculus of Variations, see Airy's Mathematical Tracts, p. 233.
of geometers. It is the path described by a particle of atmosphere in a Cyclone. And as the subject illustrates various portions of our treatise, and likewise has an interest for all whose friends or goods have ever to double the Cape of Good Hope, or to cross the Indian Ocean, I venture to say a few words on the general subject of Revolving Storms. e The definition I take to be as follows:
A Cyclone is an elliptic or circular mass of atmosphere revolving around an axis which itself describes an orbit approximately parabolic (3.)
A Cyclonoid is the locus of a point revolving at a given distance around an axis, which itself moves in a parabolic orbit. (Compare chapters xiii. and xxii.)
This definition gives three points for consideration, 1, the rotatory
(e)-It will be understood, of course, that I speak in these matters simply as a landsman. All that I know of them is derived from a perusal of Piddington's work, and from the kind instruction of our Captain during a voyage round the Cape, when for two or three weeks we were more or less within the storm field. In this place I may add, though they are somewhat foreign to our subject, the notes which foretell a revolving storm. They are1. A glare, or "greasy halo" round the sun; 2. A huge black bank of clouds in the direction of the centre, or "vortex"; 3. The sudden fall of the mercury in the barometer; and 4. The cyclonal wave, which perhaps is caused by the Resultant of the onward Forces on the water, and which takes the form of a heavy "sea," rising on a sudden and after continuing for three or four minutes subsiding into comparative calm until another cyclonal wave advances. When the storm has passed over any part of the ocean, its late track will generally be indicated by the furious cross-sea which it leaves in its wake. For these observations I must refer to "Remarks on Revolving Storms, published by order of the Lords Commissioners of the Admiralty" (6d.,) as I have not been so fortunate as to meet with Piddington's work in Melbourne.
motion of the mass; 2, the parabolic motion of the axis; and 3, the motion of any particle in the cyclone which results from the combination of the former two.
I. Since Cyclones revolve in opposite directions in the northern and southern hemispheres, let us take one in the latter, and consider it of course to revolve from left to right, that is, in the same direction as the hands of a watch. And let us conceive that the axis is stationary, and therefore the only motion is that of rotation.
Then, in the first place we may discover the position or bearing of the centre, or vortex, of the storm, with regard to the vessel. For since the motion of the atmospheric mass is circular, the direction of it, at any point, will appear, or rather be felt, along the tangent to the circle at the point. (chap. xv.) Thus, a ship at N would feel a wind from the west; a ship at E. a wind from the north, and so on (4.)
Hence it necessarily follows, since a straight line drawn at right angles to the tangent of a circle must pass through the centre (Euc. iii., 19,) that the centre must always lie in a straight line at right angles to, or in nautical language, eight points of the compass from, (ch. xvi.) the direction of the wind (4.) Thus we have the following general rule"Look to the wind's eye, set its bear
ing by the compass, and take the eight points to the LEFT (since we have conceived the storm to be in the southern hemisphere,--to the RIGHT, if in the northern.) In other words, if a man stand
facing the wind in a circular storm, the centre of the storm will be on his left hand if in the Southern Hemisphere, and on his right hand if in the Northern.
Secondly, since the wind revolves in a circle, the axis still being considered stationary, the ship of course would, if put before