604 E A K T H so that the number of seconds of time by which at the maximum the pendulum is accelerated is about half the number of seconds of angle in the maximum deflection. Principles of Calculation. Let a, a be the mutual azimuths of two points P, Q on a spheroid, it the chord line joining them, p, P- the angles made hy the chord with the normals at P and Q, <, <}> , u their latitudes and difference of longitude, and x +y +-|- a -l the equation of the surface; Q> then if the plane xz passes through P the co-ordinates of P and Q will be a x=- cos A 2/=0 z=-(l -e 2 ) sin x 7 cos <p cos w , a y=-rr cos <f> sin o> , where A = (l -e 2 sin 5 4>)i, A = (l -e 2 sin 2 </> )*, and e is the eccen tricity. Let /, g, h be the direction cosines of the normal to that plane which contains the normal at P and the point Q, and whose inclinations to the meridian plane of P is = o ; let also I, m, n and V, m , n be the direction cosines of the normal at P, and of the tangent to the surface at P which lies in the plane passing through Q, then since the first line is perpendicular to each of the other two and to the chord k, whose direction cosines are proportional to x - x , y" - y , z -z, we have these three equations fl +gm +hn = fl +gm + hn . Eliminate f, g, h from these equations, and substitute I = cos <p I = - sin (f> cos a m = m = sin a n = sin </> = cos <J> cos o , and we get (x f - a;) sin <}> + y cot a - (z 1 - z) cos <f> = . The substitution of the values of x, z, x , t/, z in this equation will give immediately the value of cot a ; and if we put f for the corresponding azimuths on a sphere, or on the supposition 6 = 0, the following relations exist cot a - cot = e 2 C _2L5L i cos <f> A , , , f , cos * Q cot a - cot C -e 2 1- -S, cos 4> A A sin (p - A sin <p = sin w Q . If from Q we let fall a perpendicular on the meridian plane of P, and from P let fall a perpendicular on the meridian plane of Q, then the following equations become geometrically evident : k sin p. sin a = , cos <f> sin u k sin n sin a = - cos <j> sin co . Now in any surface u = we have - COS /X - COS /i = . , . du , . .du . . , du (x - x) -v- + (t/ - y) -T- + & - 2) -T- e&e y/ dy ^ dz /du? du? dw a V<fc s dy* + <& du . , .du du , dy * du in the present case, if we put then COS/i=rAU; COS /x = r^ Let M b such an angle that (1 - e 2 )^ sin $> = A sin u cos = A cos u , then on expressing x, x , z, z 1 in terms of u and u , U = 1 cos u cos u cos u - sin w sin v! also, if v be the third side of a spherical triangle, of which two sides are TT - u and ^ir - w and the included angle o>, using i subsidiary angle >// such that v . u -u u + u sin = e sin - cos M m A we obtain finally the following equations : k = 2a cos sin A cos /i = A sec cos ^ = A sec
sin A sin a
sin fj. sin o = cos u sin . , sin a = cos u sm a> . These determine rigorously the distance, and the mutual zenith distances and azimuths, of any two points on a spheroid whose latitudes and difference of longitude are given. By a series of reductions from the equations containing , f it may be shown that where < is the mean of (f> and <f> , and the higher powers of e are neglected. A short computation will show that the small quantity on the right-hand side of this equation can never amount even to the ten thousandth part of a second, which is, practically speaking, zero ; consequently the sum of the azimuths a + a on the spheroid is equal to the sum of the spherical azimuths, whence follows this very important theorem (known as Dalby s theorem). If <p, $ be the latitudes of two points on the surface of a spheroid, u their difference of longitude, o, o their reciprocal azimuths, , w tan - = - cot a + a The vertical plane at P passing through Q and the vertical plane at Q passing through P cut the surface of the spheroid in two dis tinct curves. The greatest distance apart of these curves is, if a = the mean azimuth of PQ, sin 2a . This is a very small quantity ; for even in the case of a line of 100 miles in length having a mean azimuth o = 45 in the latitude of Great Britain, it will only amount to half an inch, whilst for a line of fifty miles it cannot exceed the sixteenth part of an inch. The geodesic line joining P and Q lies wholly between these two curves. 1 If we designate by P , Q the two curves (the former being that in the vertical plane through P), then, neglecting quantities of the order e"(P, where 6 is the angular distance of P and Q at the centre of the earth, the geodesic curve makes with P at P an angle equal to the angle it makes with Q at Q, each of these angles being a third of the angle of intersection of P and Q . The difference of length of the geodesic line and either of the curves P , Q is, s being the length of either, 360 4 e 4 cos 4 (J> sin 2 2a At least this is an approximate expression. Supposing the angle PQ to be as much as 10, this quantity would be less than one hundredth of an inch. An idea of the course of a geodesic line may be gathered from the following example. Let the line be that joining Cadiz and St Petersburg, whose approximate positions are Cadiz. St Petersburg. Lat. 36 22 N 59 56 N. Long. 6 18 w 30 17 K If G be the point on the geodesic corresponding to F on that one of the plane curves which contains the normal at Cadiz (by "corresponding" we mean that F and G are on a meridian) then G is to the north of F ; at a quarter of the whole distance from Cadiz GF is 458 feet, at half the dis- 1 See a paper On the course of Geodesic Lines on the Earth s
Surface" in the Philosophical Magazine for 1870.
