" CHAPTER II. LAPLACE'S COEFFICIENTS AND FUNCTIONS. 17. IN the present Chapter we shall develop the properties of those remarkable quantities which have received the name of their great discoverer, under the designation of LAPLACE'S COEFFICIENTS AND FUNCTIONS. To do this it will be necessary to anticipate the subject of the following Chapter, and to bring in here a Proposition which should properly stand at the head of that division of this treatise. PROP. To obtain formula for the calculation of the attraction of a heterogeneous mass upon any particle. 18. Let p be the density of the body at the point (xyz); fgh the co-ordinates of the attracted particle; and, as before, suppose that A, B, C are the attractions parallel to the Then the limits being determined by the equation to the surface of the body. p dx dy dz {( ƒ — x)2 + (g − y)2 + (k − 2)2 } ↓ 3 19. It follows, then, that the calculation of the attractions A, B, C depends upon that of V. This function cannot be FORMULE FOR HETEROGENEOUS MASS. POTENTIAL. 15 calculated except when expanded into a series. It is a function of great importance in Physics: and, for the sake of a name, has been denominated the Potential of the attracting mass, as upon its value the amount of the attractive force of the body depends. 20. As the axes and origin of co-ordinates in the previous Article are altogether arbitrary, it follows that if r be the distance of the attracted point from any fixed point in the attracting body, then the attraction in the line of r, towards dV the origin of r, = dr df2 dg2 dh = 0, or — 4πр', αc cording as the attracted particle is not or is part of the mass itself; p' being the density of the attracted particle in the latter case. 21. By differentiating V, we have f 'p {2 (ƒ— x)2 — ( g − y)2 — (h — z)"} dx dy dz 2 'p {2 (g − y)2 — ( ƒ — x)2 — (h — z)2} dx dy dz {(f−x)*+ (g− y)2 + (h — z)* } * `p {2 (h — 2)2 — ( ƒ— x)2 — (9 − y)2} dx dy dz ; {(ƒ − x)2 + (g − y)2 + (h − z)*} § When the attracted particle is not a portion of the attracting mass itself, then xyz will never equal fgh respectively, and consequently the expression under the signs of integration vanishes for every particle of the mass : This equation was first given by Laplace and Poisson was the first who showed that it is not true when the attracted particle is part of the attracting mass. In that case the denominator of the fraction under the signs of integration vanishes, 0 and the fraction becomes, when x = ƒ, y=g, z=h. d2V d2V d2 V To determine the value of df2 dg2 +- in that case, dh suppose a sphere described in the body, so that it shall include the attracted particle; and let VU+ U', U referring to the sphere, and U' to the excess of the body over the sphere. Then, by what is already proved, The centre of the sphere may be chosen as near the attracted particle as we please; and therefore the radius of the sphere may be taken so small that its density may be considered uniform and equal to that at the point (fgh), which we shall call p'. Let f'g'h' be the co-ordinates to the centre of the sphere; then the attractions of the sphere on the attracted point parallel to the axes are, by Art. 3, when the attracted particle is within the attracting mass. 22. It may be shown by precisely the same process as in the previous Article, that where R= {(ƒ— x)2 + (g − y)2 + (k − z)*} ̄*, the reciprocal of the distance of any point of the body from the attracted particle. PROP. To transform the partial differential equation in R into polar co-ordinates. 23. Let row be the co-ordinates of (fgh), and r'e'w' of (xyz), the angles 0 and ' being measured from the axis of z; and w' being the angles which the planes on which 0 and ' are measured make with the plane zx. Then hr cos 0, g'r' sin e' sin w', h' r' cos 0'. h √ ƒTM2 + g2 + h2 dR dr dR de tan @= > g dR do + + dr df do df dw dƒ' d dR de d dR do + + df dr df df do df df dw df d dR dr |