82. "If the parabolic orbits of two comets intersect the circular orbit of the earth in the same two points, then if t1 and to be the times in which the comets move from one point to the other, (t1 + t2)% + (t.1 a year being the unit of time.” SOLUTION BY WALTER SIVERLY, OIL CITY, PA. Let c represent the length of the chord of the earth's orbit joining the two points. It is shown in works on Central Forces that, PARADOX. BY G. SHAW, KEMBLE, ONT., CANADA.-Suppose a = b. Multiply both sides of this equality by b, and we have ab= b2. Subtract a2 from each side of this equation and we have ab- a2 — b2 — a2 ... (1) By factoring (1) we have (b a) (ba). Divide this eqn. a)a = a = a + a = 2a. = [The fallacy in the above obviously consists in considering a and b as like For, in the first member of (1) because a=b, a = O, and in the second member we have the Now when a and b are unlike the only two fac and unlike at the same time. b a. zero, b and a + b, neither of which can be but when a2, the only two factors of which are a a2 = a2 α and a; the first of these two factors, aa, being zero, the other factor a, may be repeated as often as we please without changing the value of the product.-Ed] PROBLEMS. 83. BY GEO. L. DAKE, CLEVELAND, OHIO.-A point is given within two lines which form a given angle with one another. Required the shortest line which can be drawn through this point, terminated by the given lines. 84. BY PHILLIP HOGLAN, NEWCOMERSTOWN, OHIO.-The centres of two spheres whose radii are 12 ft. and 5 ft., respectively, are at opposite extremities of the diameter of a circle of 13 ft. radius. Find a point in the cirumference of this circle from which the greatest portion of spherical surface is visible. 85. BY PROF. JAMES G. CLARK, LIBERTY, Mo.- In a quadrilateral there are given, the length and position of the lower base, the lengths of the two sides, the length of the upper base and the position of a point through which it passes required to construct the quadrilateral. 86. BY PROF. J. S. HAYES, HODGENVILLE, KY.- Prove that the attraction of a sphere of uniform density upon an external point is the same as if all the matter of the sphere were concentrated at its centre. 87. By G. M. DAY, LOCKPORT, N. Y.-There are n tickets in a bag numbered 1, 2, 3 ... n. A man draws three tickets together at random and is to receive a number of shillings equal to the product of the numbers he draws. Find the value of his expectation. 88. BY PROF. H. T. J. LUDWICK, SALISBURY, N. C.- An ellipse revolves about its latus rectum; show that the volumes of the solids generated by the larger and smaller segments are respectively equal to 2 Απα 3 (1-e2) 3e and 89. BY ARTEMAS MARTIN, ERIE, PA.-A sphere, radius r, rolls down the surface of another sphere of the same material, radius R, placed on a horizontal plane. The surfaces of both spheres and plane are rough enough to secure perfect rolling. Determine the motion of the spheres, the point of separation and the equation of the curve described by the center of the upper sphere. 90. BY R. J. ADCOCK, MONMOUTH, ILL.- Let an oblate ellipsoid of revolution of homogeneous density rotate about one of its greatest diameters. What must be the ratio of its axes, that a column of liquid along the greatest diameter at right angles to the axis of rotation may just balance one along the shortest diameter? a2÷dk, being the same as in the case of the earth, where a = angular velocity of rotation, k1 = attraction of a spherical unit of mass for another at the distance unity between centres, o mean density. 1 91. BY PROF. W. W. JOHNSON, ANNAPOLIS, MD.—Let a sphere, rotating with the angular velocity w, be provided with pivots at the extremities of a diameter inclined to the axis at the angle a. If these pivots be suddenly caught in fixed sockets, the sphere will rotate about the new axis with the rate w cos a. If the pivots be caught by a ring which is itself free to rotate about an axis perpendicular to the new axis of the sphere and passing through its centre, the rate of rotation about this axis will be w sin a. The original rotation and these component rotations represent kinetic energies which are proportional to w2, w2 cos2 a and w2 sin' a: hence there is no loss of energy and no shock. That is, every particle will, when the pivots are caught, undergo no sudden change in velocity or direction. Prove the truth of this by spherical trigonometry. Note. If solutions of the problems proposed in any No. are received by the 10th of the next succeeding month, they will in general be either published in the first succeeding No., or noticed at the head of "Solutions of Problems"; but if received after the 10th of such month, and if a solution of the problem, or problems, is published in the next No., no notice in general is given of such solution.-Ed. THE ANALYST. VOL. II. Nov., 1875. No. 6. ON THE DEVELOPMENT OF THE PERTURBATIVE BY G. W. HILL, NYACK TURNPIKE, N. Y. 1. THERE are two modes of developing this function. In one, the numerical values of the elements involved are employed from the outset, and the results obtained belong only to the special case treated. This mode has been, almost exclusively, followed by Hansen, and is, perhaps, to be recommended. when numerical results are chiefly desired. In the other, all the elements are left indeterminate, and thus is obtained a literal development. possessing as much generality as possible. Certain investigations, arising from Jacobi's treatment of dynamical equations and Delaunay's method in the lunar theory, have invested the latter mode of development with additional interest, and with it we shall be exclusively engaged in this article. In Liouville's Journal for 1860, M. Puiseux has given us two memoirs on this subject, in which appears the general term of this function, but his formulæ seem susceptible of modifications which would render them much simpler. More recently, in the volume of the same journal for 1873, M. Bourget has presented the development in a more concise form by employing the Besselian functions, but as he discards the use of the functions b, his formulæ, on this account are more complex. It is hoped, that, even if the expressions, given hereafter, are deemed too cumbrous for practical use, they may still possess some interest from a theoretical point of view. 2. It is known that if we have a function S of a variable 【, which is never infinite, and such that the relation is satisfied for all integral values of i both positive and negative, it can be developed in a series of the form 2. (K) cos i + K{3) sin i¿), in which i denotes a positive integer; and that, in the cases where this series is infinite, it is convergent. In general the handling of periodic series is easier if we introduce imaginary exponentials in the place of the circular functions. Thus, & denoting the base of natural logarithms, we shall put ze 2 cos( = z + z−1, 21 (-1) sinzz1, whence z = cos ( + √(—1) sin (, zi = cos i + 1/(-1) sini. ناچ The above theorem then comes to the same thing as to say that S is developable in a series of the form Σ. Cz, where the summation is extended to negative as well as positive values of i. The coefficients K are given in terms of the coefficients C by the equations KC + C-i, except the case where i = 0, when K = Co. It will be seen that when S is real, C; is a complex number a + b√-1, and C,, its conjugate a by—1, which renders the coefficients K real, as they should be. The integral Szide = S(cos i + √(−1) sin i¿)dę, taken between the limits 0 and 27, vanishes in all cases except when i=0. when its value is 27. Hence any function, capable of expansion in a series of positive and negative integral powers of z, integrated with respect to between these limits, gives, as the result, 27 times the coefficient of 2o in its expansion. And as the coefficient of 20 in the function Szi is evidently C1, we have This equation holds for all values of i, negative as well as positive, zero included. 3. Let us now suppose that denotes the mean anomaly of a planet, and let u be the eccentric anomaly, connected with the former by the equation, e being the eccentricity, И e sin u 5. In like manner as for, we will introduce the imaginary exponential thus, as the last equation can be written. 8=8 |