subjects are considered, it is unnecessary to do more than insert the forms of them which express the circumstances of our problem, and explain the symbols employed. A particle is supposed to be projected with a given velocity (which in the case of a falling particle may be zero) in a given direction. The place on the earth's surface, whence the particle is projected, is taken as the origin; the axes of x and of y are taken in the horizontal plane, and are respectively north and south, and east and west, the positive direction of a being taken towards the south, and that of y towards the west; and the z-axis is the vertical line measured upwards from the earth towards the zenith of the place; and this line may be assumed without sensible error to pass through the earth's centre. The latitude of the place is λ; and is the angular velocity of the earth; g is the force of gravity of the earth, and is considered to be constant for all points of the path of the particle (m). Then the equations of motion are Now is a very small quantity; to determine its value I will take a second to be the unit of time: then, as a mean sidereal day contains 86164-09 seconds, 11 2π ='00007292. Consequently 2, which enters into the preceding equations, is an extremely small fraction. Also in the present problem, notwithstanding the increase of range now obtained by the improved weapons of projection, x, y, z are all very small parts of the earth's radius; and therefore in the first approximate solution of the preceding equations, I will neglect those terms which contain products of these coordinates and of o2; so that the equations become As these are linear equations of the first order, they are easily integrated; and if u=the velocity of projection, and a, B, y are the direction-angles of the line of projection, we have x=ut cos a-u w sin à cos ẞ t2, y=ut cos ß+u∞ (cos a sinλ+cos y cos λ) {3 — wg così, z=ut cosy - (2+ + uw cosλ cos B) { ; which three equations give the place of the projectile at the time t. Now, without proceeding further at present in the process of approximation, let us consider two particular cases and results, which are of considerable interest. (1) Let the body fall, as e. g. down a mine, without any initial velocity; then u=0; cos a=cos B=0; cos y=-1; The first equation shows that there is no deviation in the line of the meridian: from the second we infer a deviation towards the east; that is, in the direction towards which the earth is moving; which varies as the cube of the time of falling; and that this deviation is greatest at the equator, where λ=0: and the last equa tion shows that the earth's rotation does not produce any alteration in the time of falling. If we eliminate t, and take z downwards to be positive, which is the equation to a semicubical parabola, and shows that the square of the deviation towards the east varies as the cube of the space through which the particle has fallen. (2) Let the particle be projected due southwards at an angle of elevation equal to 0; then From the first and the last of these equations we infer that neither the time nor the range on the meridian is altered by the rotation of the earth. Also when :=0, 2 u sin 0 that is, when the projectile strikes the ground, t= 9 ; in which case {sin cos λ+3 cos e sin λ }; and therefore the point where the projectile strikes the ground is west of the meridian so long as is less than 180° – tan-1 (3 tan λ): and the deviation vanishes if 0=180°-3 tan-1 (3 tan λ). The deviation is eastwards if is greater than 180°-3 tan 1 (3 tan λ). Now these results, which have herein been applied to the motion of a material particle, are also true of that of the centre of gravity of a body. Neglecting therefore the resistance of the air, and the action due to the rotation of a ball or bolt, they are applicable to rifle and cannon practice, and we have the following results. When the shot is fired due north or south, the range in that direction is not altered, but there is always a deviation of the shot, the value of which at the point of impact on the ground is given in the last equation. Also from the preceding equations the following results may be deduced :-When the shot is fired due east, the range eastwards is increased or diminished according as the angle of elevation of the gun is less than or greater than 60°; and the deviation is southwards for all places in the northern hemisphere, and northwards for all places in the southern hemisphere. When the shot is fired due west, the range is increased or diminished according as the angle of elevation is greater than or less than 60°; and the deviation is northwards for all places in the northern hemisphere, and southwards for all places in the southern hemisphere. So that for firing from a place in a direction coincident with the parallel of latitude, and with an elevation less than 60°, the range is increased or diminished according as we fire eastwards or westwards; and the difference between the two ranges = 8 u3 w cos λ `{3 (cos 0)2 — (sin 0)2} ; and if the place is in the northern hemisphere, the deviation parallel to the meridian is north or south, according as we fire west or east. Now these effects have been inferred from the equations of motion, simplified by the assumption that products of w2, and one of the relative coordinates of m, are small quantities, and are to be neglected. Let us now retain these quantities, and assume that products of 3 and of a small variable are to be neglected; and that all small quantities of a lower order are to be retained. We shall suppose the values of x, y, z given above to be approximate solutions of the first order of the equations; and if, according to the general method of solution adopted in such cases, we substitute these values in terms involving w2x, w2y, and w2z, that is in the smallest terms which we intend to retain, and, omitting terms of higher orders, then integrate the simultaneous differential equations thus formed, the results are x=ut cos a-u w sin λ cos ẞ t2 — u w2 sin à (cos a sinλ+cos y cos x)+go2 sin à cosa; 2 1 z=ut cosy t2-u w cos à cos ẞ t2 — u w2 cos à (cos a sinλ+cos y cos x)+gw2 (cos λ)2 t t3 2 These equations, of course, give results corresponding to particular initial circumstances. I will take only two. (1) Let the body fall without any initial velocity; then u=0, cos a=cos ß=0, cos y=−1; The first equation shows that there is a deviation of the falling particle in the line of the meridian towards the south; and the second shows that the deviation in the parallel of latitude is towards the east; so that the resulting deviation of the falling body is towards the south-east. This result is in accordance with the case many years ago investigated by Hooke, the contemporary of Sir I. Newton. From the last equation it appears that the space due to a given time is less than it would be if there were no rotation. (2) Let the body be projected due southwards at an angle of elevation equal to e, so that cos a=cos 0; cos ß=0; cos y=sin 0; then When the projectile strikes the ground, z=0; and approximately t= which case 4 u3 w (sin 0)2 { sin ◊ cosλ+3 cos ◊ sin λ } ; y= 3g2 which is the same expression as that just now interpreted. Consequently the aim of a long-range gun pointed due north or south must be in accordance with the preceding explanations. On the Calculus of Functions, with Remarks on the Theory of Electricity. By W. H. L. RUSSELL, A.B. The object of this paper was to give some account of a method discovered by the author for the solution of functional equations with rational quantities, known functions of the independent variable, as the arguments of the unknown functions. The solutions were given by series, and also in terms of definite integrals. On Petzval's Asymptotic Method of solving Differential Equations. By WILLIAM SPOTTISWOODE, M.A., F.R.S. The researches of M. Petzval here brought under notice are directed to the solu tion of those linear differential equations with variable coefficients which have reference to motions, themselves small, but propagated to great distances. In such equations y usually represents the disturbance, and r the distance from the origin. If then the solution y=f(x) be considered as the equation to a curve, the method proposed by the author will give the values of y corresponding to large values of z; in other words, the asymptotes to the curve in question. Hence the name "Asymptotic Solution." With a view to this object M. Petzval proposes the following question: Can any general laws be established, with respect to the coefficients of a differential equa tion, capable of furnishing criteria for determining the nature of the particular integrals which satisfy it? Having first made such a classification of functions as renders his conclusions capable of conversion, in the logical sense of the term, he proceeds to form, from a given equation of the degree n, Xny(")+Xn-1y(n−1) + .. Xo y= (X,y)(*)=0, the equation of the degree (n+r), (Z,z)(n+r)=0, arising from the introduction of r particular integrals of a given form. Passing over the case of algebraic integrals, some of the criteria of which are common to exponentials, the more important cases are as follow: I. Particular integrals of the form ex Q, where Q is an entire algebraic polynomial. (1) To a level (i. e. an equality of degrees among consecutive coefficients) in (X, y)(")=0, there corresponds in general a level among those of (Z,z)(n+1)=0. (2) To a level among Xk+r-1, Xk+r-2,.. Xk, followed by a continuous fall among X-1, Xk-2,.. Xo, of (X, y)(")=0, there corresponds a level among Zk+r, Zk+r−1, . . Zk, followed by a similar fall among Zk-1, Zk-2, .. Zo, of (Z, z)(n+1)=0. II. Of the form ax2+4(x) Q, or eßx Q, where (x) is defined to belong to the author's first class. (1) To a continuous rise among Xu, Xn−1,.. Xn-r+1, of (X, y)(")=0, there corresponds in general a similar rise among Zn, Zn-1,.. Zn-r, of (Z,z) (a+1)=0. (2) To a continuous fall among Xx-1, Xk-2,.. Xo, of (X,y)(")=0, there corresponds in general a similar fall among Zk-1, Zk-2,.. Zo, of (Z, z)(n+1)=0. III. Of the form (z)dz Q, where the degree of f(x)dx is fractional, 2, and consequently that of p(x) is P—9— — §. = (1) If be a proper positive fraction, to a level among Xx−1, Xk−2, .... Xô, of (x,y)()=0, there corresponds a fall among Zr-1,Zr-2,.. Zo, of (Z,=) (+r)=0, amounting in all to νδ (2) If? be an improper positive fraction, to a level among Xn, Xa-1,.. Xn-r+1, of (x, y)(n)=0, there corresponds a rise among Zn+r, Za+r-1. to a series of coefficients Xn, Xn-1, . . Xn-r+1, of (X, y)(»)=0, free from the factor (-a), there corresponds a series (-a) a+b+.. Zn+r (x − a)3 + ·· Zm+r−1, . . (x− a)k Zn+1, of (Z, z)n+r=0. The general result to which the author brings these conclusions, together with the exceptional cases, not here specified, will be best exhibited by the following examples. Example 1. Let the degrees of the coefficients be 1, 3, 4, 4, 4, 3, 2, 1, 0, the equation being of course of the 8th degree. Then construct the following figure, in which the ordinates are proportional to the degrees of the coefficients : The differences between the degrees of the coefficients are 2, 1, 0, 0, -1, -1, −1, −1; and consequently the degrees of (x) in the particular integrals of the equation will be 3, 2, 1, 1, 0, 0, 0, 0; so that in the general solution there will be One integral of the form a3+6x+yæQ, Four of a purely algebraic form. Example 2. Let the degrees of the coefficients be 1, 3, 3, 4, 4, 0, 2, 3, 2, 0, the equation being of the 9th degree. Then form the figure where, after bridging over the re-entering angles, the differences of the degrees are About the degree -1 there is a difficulty; but the author suggests that the negative index arises from an accidental cancelling of the highest power of x in Zo, and that it may probably be replaced by zero. On the Reduction of the decadic Binary Quantic to its Canonical Form. Professor Sylvester has shown that the quantic in which A is a constant, and V a covariant of the product u,,,,.. un, satisfying a certain differential equation. In applying his method to the 10th degree, the greatest |