48 INFINITESIMAL CALCULUS This result can be readily represented as a theorem in double integration, as follows. If the double integral be taken for all positive values of x and y subject to the condition x + y<a, its value is represented by For, considering x as constant, and integrating with respect to y between the limits and a - x, the value of the double integral becomes 157. The preceding result, first given by Euler, was generalized by Dirichlet (Liouville s Journal, 1839), and extended to a large class of multiple integrals by the following theorem. Let V= fff. -i . . . dxdyih in which the variables x, y, z, &c. , are always positive, and subject only to the condition x + y + z. . . <!,. Y(l)T(m)Y(n} . . . V = r then It will be sufficient here to show that the theorem is true in the case of three variables, x, y, z ; i.e., let V = /// ~ ] y m ~ 1 ~" ~ l dxdydz , subject to the foregoing conditions. Integrating with respect to z, considering a; and y constants we get V = ffa - V - l (l-x- yYdxdy , in which x and y are positive, and subject to the condition x + y<l. If we next integrate with respect to y, between the limits and 1 - x, we have /*!-* Tfin Accordingly T(m+n + li/ Q T(i)Y(n) T(l)T(m)T(n) 158. The preceding theorem when extended to n variables can be stated somewhat more generally, viz., if a, j8, y, . . . p, q, r, . . . being positive quantities, then we shall have r i + + + . . . V P 1 r This readily follows from the preceding by making In the case of three variables this theorem contains a large num ber of results relative to volumes, centres of gravity, moments of inertia, &c. The remarkable elegance and generality of Dirichlet s theorem immediately attracted notice, and his results were speedily extended by Liouville, Catalan, Leslie Ellis, and other mathematicians of distinction. Of the results thus established we shall content our selves with giving Liouville s extension of Dirichlet s theorem (Liouville s Journal, 1839). If where x, y, z are always positive and subject to the condition II 7 Y+ ... < h , 2>qr . . . j i P f i This follows without difficulty from the preceding by assumin, ^ =2/ , . . . , and then making x + y + z. . . = u. subject to the condition x + x; + xf. . . . + x?i < R 2 , (2) The value of r( i + fff- extended to all positive values of the variables for which the expression is real, is n+l (3) The value of ffy* - 1 ij - k c*+udxdy extended to all positive values for which x + y< h is -V (*-!) Sill /ITT (4) The value of for all real values of the expression, x and y being positive, is (5) The value of extended to all positive values ofx, y, z contained within the ellipsoid ^-,+ y- + ~ = l a- b- c 2 2/V2/V2 + 1 (6) Prove that rrr f^ x i+ a ^-^- n " x "- r , (b . clx III .- 1, ; , >Ci*a CWn , JJJ V 1 - X l -X: - when extended to all values subject to the condition is equal to r where Jc=/a { + al+ . . . +a%. 159. We shall next give a short account of Legendre s formula for the calculation of log F(l +x). Adopting Gauss s definition, substituting a + l for n, and taking the logarithms of both sides of the equation of 155, we get logr(a?+l) -lim. log M - l If now a; lie between +1 and -1, we may substitute their well- / ,r , / x known expansions for log ( 1 + , j > l g ( 1 + ^ I Hence, representing the indefinite series 1 1 1
