2 2 -10 putting ra (1+ y) = a(1+ Y + Y1+...+Y+...), and observing that VI-μ2 cos w, √1-μsin w, and μ satisfy V1 Laplace's Equation, and are of the first order, we have by Art. 26, But Y1, being a function of μ, √1-μ3 cos ∞, and √1-μ3sin w of the first order, is of the form A√1-μ3 cos w+B√1 — μ2 sin w + Cμ; Hence if we take the origin of co-ordinates at the centre of gravity and therefore = 0, y = 0, z=0, we have A = 0, B=0, C=0, and therefore Y, 0, as stated in the enunciation. = PROP. To find the attraction of a heterogeneous body upon a particle without it; the body consisting of thin strata nearly spherical, homogeneous in themselves, but differing one from another in density. 45. Let a' (1+y') be the radius of the external surface of any stratum, a' being chosen so that y' = Y,'+ Y2+ ... + Y + ... (Art. 44). Since the strata are supposed not to be similar to one another, y' is a function of a' as well as of μ' and w'. Let p' be the density of the stratum of which the mean radius is a'. Now the value of V for this stratum equals the difference between the values of V for two homogeneous bodies of the density p' and mean radii a' and a-da'. But for the body of which the mean radius is a' (Art. 42) Hence for the stratum of which the external mean radius From which the attraction is easily deduced. PROP. To find the attraction of the same body on an internal particle. 46. Let r = a(1+y) be the radius of the stratum in which the attracted particle lies. Then for the strata within the surface of which the radius is a (1+y), we have But for a stratum external to the particle we have by Art. 43, a дов + da' 3 da' Y a'i-2 Y+. == S 0 ρ 3r d (ra' + 4π [ " p' {a'da ' + 2 (m2 X';' + ... + (2641) a++ X'; + ...)} da'. From this the attraction is readily obtained by differ entiating with respect to r. CHAPTER IV. ATTRACTION OF BODIES NEITHER SPHERICAL NOR SPHEROIDAL, NOR NEARLY SO. 47. THE methods which have hitherto been given enable us to find the attraction of the Earth and other bodies of our system considered as a whole. But, taking the Earth as our example, the surface is irregular, and neither exactly spherical nor spheroidal. We ought, therefore, to be able to calculate the effect of these irregularities, and with this view the present Chapter is added to what has gone before. High Table-lands may very materially affect the position of the plumb-line in some places. Enormous irregular mountain masses, like the Himmalayas, may do the same. Their effect ought, therefore, to be carefully estimated, as all instruments which are fixed by the plumb-line or spirit-level must be affected by such irregularities. PROP. To find the attraction of a slender prism of matter on a point in the line drawn to one of its extremities. B 48. Let AB be the prism, C the attracted point, P any element of the prism, AP=r, M the mass and 7 the length of the prism, AC=a, BC=b, PC=y, angle PAC= 0. As this is symmetrical with respect to a and b, it shows that the particle is attracted equally towards the two extremities of the prism; and that therefore the resultant attraction acts in a line bisecting the angle which the prism subtends at the attracted point. PROP. To find the attraction of a slender pyramid of any form upon a particle at its vertex; and also of a frustum of the pyramid. 49. Let be the length of the pyramid, a the area of a transverse section at distance unity from the vertex; r the distance of any section; ar2 is its area; p the density of the matter: then ar2pdr is the mass of an element of the pyramid, and this divided by 2 is its attraction; .. attraction of pyramid on vertex = = ['apdr = apl. 0 If d is the length of any frustum of the pyramid, and l=l'+d, then attraction of pyramid, length l', = a.pl'; .. attraction of frustum = apd. It is observable that this is quite independent of the distance of the frustum from the vertex; and therefore all portions of the pyramid of equal length, any where selected, attract the vertex equally. 50. COR. Suppose the angular width of the pyramid to be B and to remain constant, while the angular depth varies; and let k be the linear depth of the transverse section of the base; then aßlk is the area of the base; and the attraction of the whole pyramid on the vertex = pßk. Hence, all pyramids having the same angular width and the same linear depth at the base attract their vertex alike, whatever their lengths be. PROP. To find the attraction of an extensive circular plain of given depth or thickness upon a station above its middle point. 51. Let t be the thickness or depth; h the height of the particle from the nearer surface, c the radius, r the radius of any intermediate elementary annulus of the attracting mass, z its depth. The several elements of this annulus of matter will attract the particle towards the plain equally. Hence attraction of the particle 52. If the plain be of infinite extent, the attraction equals 2πpt; and this remarkable result is true, that it is independent of the distance from the plain. The same will be the case if the height of the station above the middle of the attracting |