described in the time dt, and then A M= ds = the space described in the same time. The arc AB=rdp. But the triangle AB M may be regarded as rectilinear, and right = angled at B; and if we put the angle AMB = &, we shall have A B AM sin AMB; or rdq=ds sinɛ. Multiplying by r, we have 2dq=cdt=rds sine; or c=r D, 8 sin & = rv sin ε (57). When rr1 and v= V, let ɛ = &1; we shall have cr1 V sin &1 (58). That is, when the initial velocity and radius vector are the same, the area described in the unit of time is dependent on the angle which the direction of the initial velocity makes with the radius vector; and since the eccentricity is dependent upon the value of c, (51) it follows, that when r1, V, and consequently h, are the same, that the eccentricity is dependent upon the same angle, because m is constant for the same body. We also see (53) that the semi-conjugate axis is dependent upon the same angle. But the semi-transverse axis (54) is not dependent on c. 19. Since c is twice the area described in the unit of time, in the ellipse it is evidently equal to twice the area of the ellipse divided by the time of revolution in the ellipse, which call T, and we shall have therefore, m = c2 ..m= 4 n2 a3 (59). Since m is constant for the a (1— e2)* same body, we see that the time of revolution depends wholly upon the value of the transverse axis; and if a circle and any number of ellipses be described on the same transverse axis, the time of revolution in all of them will be the same. 20. We have seen, Art 14, that when h= Calling r the radius, we have for the velocity in a circle. Calling V, the velocity in a circle, and V, the velocity in a parabola, we shall have, from (56) and (60), That is, The velocity in a parabola, at any point, is equal to the velocity in a circle passing through the same point, into the square root of 2, the centre of the circle being at the focus of the parabola. We can, therefore, find the velocity in a parabola at any point, by knowing its distance from the sun. 21. We have represented the attractive force by R=. To interpret this expression, let M equal the attraction of the central body, at the unit of distance, and then, according to NEWTON's law, M will equal the attraction at the distance r. Let my equal the attraction of the revolving body at the unit of distance, then will equal the attraction at the same distance r. But both these forces tend to draw the bodies towards each other; and hence, the whole attractive force which causes the bodies to approach each other is 22. We have already found m= T2 For any other body re volving at the mean distance d', we have m' = these two equations, we have the proportion (64) m T2: m' T'2 :: a3 : a'3. When m and m' are nearly equal, as is the case in the solar system, we may regard them equal, and we shall have That is, the squares of the periodic times are to each other as the cubes of the mean distances from the sun. This is KEPLER'S celebrated third law. From (64) we see that it is not strictly true. We have +m1 m Calling M= 1, and (+) = (+)=1+, by de M+ 3 veloping and retaining only the first powers of m, and m2, the ratio of the mean distances, or 23. When the times of revolution and the mean distances are given, we can find the difference between the masses of the two planets; where my and m2 are the masses according to NEWTON'S law. 24. Equations (59) can be employed to find the mass of a planet which has a satellite. Call t and a the time of revolution and mean distance of the satellite, m, its mass, and m, the mass of the planet. We obtain All the quantities a, a, T, t, being derived from observation, m, is known, which substituted in (69) will give the value of mp. ); or m, = M — 4 n2 ( — — a2). ON SPHERICAL ANALYSIS.* By GEORGE EASTWOOD, Saxonville, Mass. PROPOSITION I. To find the equation of a great circle of the sphere, which passes through the origin, and which makes a given angle with the axis of x. Let OX, OY, be quadrantal arcs of great circles of the sphere intersecting each other in the point 0; and let OP be the great circle whose equation is required. Through any point P, pass the great circle arcs XP M, YPN, and designate ON by x, OM by y, OP by r, the angle YOX by w, and the angle POX by q. Then, in the spherical triangle X0 M, we have *The following are among the works and memoirs I have had occasion to consult. -Grundriss der Sphärick Analytischen, by Professor GUDERMANN, of Cleve, published about 1830, in which, so far as I know, the idea of expressing the equations of spherical curves in tangent-functions of their rectangular co-ordinate arcs, was first conceived and developed into a system. Two elaborate memoirs On the Equations of Loci traced on the Surface of the Sphere, by the late Professor DAVIES, of the Royal Military Acade cos XM = cos 0 M cos 0 X + sin 0 M sin O X cos ≤ 0, = = siny cosa, since 0X a quadrant, cos O M = cos y = cos XM cos X0+ sin XM sin XO cos <X, my, Woolwich, England, published in 1832 in the XIIth Volume of the Transactions of the Royal Society of Edinburgh." Researches on Spherical Geometry," by the late Professor GILL, published in 1836, in the Mathematical Miscellany. - Two Geometrical Memoirs On the General Properties of the Cone of the Second Degree, and on Spherical Conics, by CHASLES; translated by Rev. CHARLES GRAVES, Fellow and Tutor of Trinity College, Dublin, with an Appendix On the Application of Analysis to Spherical Geometry, and published in 1841.- Essai de Géomètrie Analytique de la Sphère, by M. BORGNET, published by BACHELIER in 1847. I only know of this work through references to it, all my efforts to procure a copy having been unsuccessful. -Memoir On Spherical Analysis, by M. VANNSON, published in TERQUEM and GERONO'S Nouvelles Annales de Mathématiques, Tome XVII. |