Laplace's equation in generalized co-or dinates. we find for the symbol 2 in terms of the generalized co-ordinates d + R′′ (cos a cos a' — cos a') de Case of rectangular co-ordinates, curved or plane. d + R' (cos a" cos a - cos a') de" αξ d de + R (cos a' cos a′′ – cos a) ] } ...(12), where for Q, its value by (8) in terms of a, a', a" is to be used, and a, a', a", R, R', R' are all known functions of έ, ¿', έ" when the system of co-ordinates is completely defined. (f) For the case of rectangular co-ordinates whether plane or curved a = a' = a" = A = A' = A" = 90° and Q = 1, and therefore we have which is the formula originally given by Lamé for expressing in terms of his orthogonal curved co-ordinate system the Fourier equations of the conduction of heat. The proof of the more general formula (12) given above is an extension, in purely analytical form, of a demonstration of Lamé's formula (13) which was given in terms relating to thermal conduction in an article "On the equations of Motion of Heat referred to curvilinear co-ordinates" in the Cambridge Mathematical Journal (1843). (g) For the particular case of polar co-ordinates, r, 0, 4, considering the rectangular parallelepiped corresponding to dr, 80, 8 we see in a moment that the lengths of its edges are dr, rde, r sin 08p. Hence in the preceding notation R = 1, R=r, =r sin 0, and Lamé's formula (13) gives R" equation in co-or 165 (h) Again let the co-ordinates be of the kind which has Laplace's been called "columnar"; that is to say, distance from an columnar axis (r), angle from a plane of reference through this axis to dinates. a plane through the axis and the specified point (ø), and distance from a plane of reference perpendicular to the axis (). The co-ordinate surfaces here are coaxal circular cylinders (r= const.), planes through the axis (const.), planes perpendicular to the axis (z = const.). The three edges of the infinitesimal rectangular parallelepiped are now dr, rdo, and dz. Hence R=1, R'=r, R' = 1, and Lamé's formula gives which is very useful for many physical problems, such as the (i) For plane rectangular co-ordinates we have R = R' = R"; Algebraic so in this case (13) becomes (with x, y, z for έ, έ, E'), (j) Suppose now it be desired to pass from plane rectangular co-ordinates to the generalized co-ordinates. Let x, y, z be expressed as functions of έ, &, "; then putting for brevity R = √(X2 + Y2 + Z3), R' = √(X"+Y" + Z'), transformation from plane rectangular to generalized coordinates. Algebraic transformation from plane rectangular to generalized coordinates. and the direction cosines of the three edges of the infinitesimal parallelepiped corresponding to dέ, dέ, dέ" are (k) It is important to remark that when these expressions for cos a, cos a', cos a", R, R', R", in terms of X, &c. are used in (8), Q becomes a complete square, so that QRR'R' is a rational homogeneous function of the 3rd degree of X, Y, Z, X', &c. For the ordinary process of finding from the direction cosines (20) of three lines, the sine of the angle between one of them and the plane of the other two gives X, Y, Z sin p= X, Y, Z ÷ RR'R" sin a.. .(21); X", Y", Z" from this and (7) we see that QRR'R" is equal to the determinant. From this and (8) we see that Square of a (X+Y2 + Z3) (X”2 + Y12 + Z'3) (X''2 + Y''2 + Z"') determi nant. -(X3+Y3+Z3)(X'X"+Y'Y"+Z'Z")3-(X"2+Y'2+Z")(X′′X+Y"Y+Z"Z)' - (X"2+Y"2+Z"2)(XX' + YY' + ZZ')' + 2 (X'X"+Y'Y" +Z'Z")(X"X+Y"Y+Z"Z)(XX'+YY'+ZZ') an algebraic identity which may be verified by expanding both members and comparing. (7) Denoting now by T the complete determinant, we have Q = Τ ..(23), and using this for Q in (12) we have a formula for in which only rational functions of X, Y, Z, X', &c. appear, and which is readily verified by comparing with the following derived from Algebraic (16) by direct transformation. (m) Go back to (18) and resolve for 8, 8, 8". We find transformation from plane rectangular to generalized coordinates. L=Y'Z'-Y''Z', M=Z'X"-Z"X', N=X'Y"-X"Y', .(24). L" d 2 M d = + T dέTM T dg + + T de T de T de" d &c. with those of the corresponding terms of (12) with (21) de' and (23) we find the two formulas, (12) and (25), identical. N" d\ + Tag.) .(25). A.-EXTENSION OF GREEN'S THEOREM. It is convenient that we should here give the demonstration of a few theorems of pure analysis, of which we shall have many and most important applications, not only in the subject of spherical harmonics, which follows immediately, but in the general theories of attraction, of fluid motion, and of the conduction of heat, and in the most practical investigations regarding electricity, and magnetic and electro-magnetic force. (a) Let U and U' denote two functions of three independent variables, x, y, z, which we may conveniently regard as rectangular co-ordinates of a point P, and let a denote a quantity which may be either constant, or any arbitrary function of the a constant gives a theorem of Green's, variables. Let fffdxdydz denote integration throughout a finite singly continuous space bounded by a close surface S; let sƒdS denote integration over the whole surface S; and let 8, prefixed to any function, denote its rate of variation at any point of S, per unit of length in the direction perpendicular to S outwards. For, taking one term of the first member alone, and integrating d(asdy) + + dxdydz dy dz ..(1). the first integral being between limits corresponding to the surface S; that is to say, being from the negative to the positive end of the portion within S, or of each portion within S, of the line x through the point (0, y, z). Now if A, and A, denote the inclination of the outward normal of the surface to this line, at points where it enters and emerges from S respectively, and if ds, and dS, denote the elements of the surface in which it is cut at these points by the rectangular prism standing on dydz, we have Thus the first integral, between the proper limits, involves the elements Ua2. cos AdS,, and - U'a2 dU dx cos AdS; the latter 2 of which, as corresponding to the lower limit, is subtracted. Hence, there being in the whole of S an element dS ̧ for each element dS,, the first integral is simply 2 |