Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/68

58 INFINITESI M A L C A L C U L IT S where du. dF du n dF_ 2 dFs (ZF_2 du l du 2 diin dF n dFn dF n di^ du. 2 dii n Accordingly the multiple integral transforms into dF, dF n ^2 The limits in the transformed integral are determined by aid of the equations which give the limits in the original. 186. Wo conclude this short account of multiple integrals with a notice of the very general and remarkable theorems relative to integrals extended through a closed surface first given by Green (Essay on the Application of Mathematics to Electricity and Magnet ism, Nottingham, 1828). Let U, V denote two functions of the rectangular coordinates x, y, z which are finite, and have their first differential coefficients finite, for all points within a closed surface ; then, since dU dV d*V we have dx J dx /7Trf/ TT rZV 7 /r 7 / TT /// -3- U-y- }dxdydz = // dydtSl TJ, JJJ dx dx ) JJ 2 the integrals being extended to all points within the surface. Also, since the bounding surface is closed, any right line which meets the bounding surface, must cut it in an even number of points ; hence the integral dV dV |is - U a 2p where x. 2 , x jt U 2 , Uj, &c., are the values of x, &c. , for two corre sponding points of intersection with the boundary by the infinitely thin cylinder standing on dydz, and by 2 is denoted the summation taken for all such values. Again, if rt*S. 2 , dS^ be the corresponding elements of surface, and a 2 , a l the angles which the exterior normal to the surface at each of these points makes with the positive direction of the axis of x, we have dydz = cos a. 2 dS. 2 = - cos oj dS 1 ; hence it is readily seen that the integral is equal to taken for every element of the boundary, whether it consist of one closed surface or of several. Accordingly, we get J 7 f/7~ flUd Vj /7~rT ft7V 9* + JJJ ^ dx- dXdydZ =M U ^x in which the former integrals are taken for every point within any space, and the latter integral taken for each point on the boundary of that space. This may be written du dv . rr dy /7r TT ^ 2 v , , , ^^ **Ay**-JJV -^ cos a rf3 -JJjM -^ dxdyd: . Taking the corresponding equations relative to y and z, we* have by addition, //(I dU_ dV_ dx dx dy dy dV dV -T- cos a + - r ~ cos dx dy dV ~dz~ Again, if dn be th? clement of the normal, measured outwards, at the element f/S, we readily get dx dn dy du dV dV . cos a + ; cos ax dy cos 7= - dV T- COS y = dz dV -y dn dU dV dJ dV, , . ~; --- / --- ^ ~5 -- T- dxdydz dy dy dz dz J /7"rr dy ? a /7T^i (py , d ~V d ~Vi 7 = // U -j- dS - /// U -j-r + -j-s + ^ n; dxdydz JJ dn JJJ dx* dy 2 dz 2 J ffv du /Q /77Y (PU , {p u d-u 7 , . // v ~^ da - /// v ^ + . + __ _ dxdydz . JJ dti JJJ dx* dy 2 dz- j The latter equation is obtained by an interchange of U and V. This is Green s fundamental theorem, in the case where U and V are continuous functions. 187. The modification when one of the functions, U for example, becomes infinite for a point within the surface was also investigated by Green. Suppose this to happen at one point, P, only; moreover, infinitely near to P let U be sensibly = , r being the distance from P. Next suppose an indefinitely small sphere, of radius a, described with P as centre. Then it is clear that Green s equation holds for all the space exterior to this sphere. Also, since the triple integrals may be extended throughout the entire space. Moreover, the part of //U- dS due to the surface of the sphere JJ dn is plainly infinitely small of the order of rt. It only remains to con sider the value of /7v -=dS taken over the surface of the sphere. But, since { ~ = - , this becomes - 4^!, where V, is the value dn a- of V at the point P. Hence, denoting we have zVvJJ -J where, as before, the double integrals are extended over the boun ding surface or surfaces, and the triple integrals taken through out the entire space enclosed. These theorems of Green have been generalized by Sir ~Y. Thomson ; thus, if a be another continuous function of x, y, z, we get, by a similar treatment, instead of Green s first equations, dV r~ dx dU dV ~7 -- J dy dy dJ dV /7""V dU 7Q /77V I (l ( dlj (l ( o dU =// a-V -j S - /// V ] -j-j o 2 -j + -r- er-r- JJ dn JJJ dx dx J dy dy ) /Tzn dV 7 /77"rT d ( f/v ^ d I o (lv // a^U -= aa - /// U -^- a 2 -3 + -r- a 2 -j I JJ dn JJJ dx dx J dy dy ) U dV dxdydz , with a corresponding modification when one of the functions becomes infinite at one or more interior points. In the case of many-valued functions, another modification of Green s theorem was established by Helmholtz (" Ueber Integral e der Hydrodynamischen Gleichungen welche den Wirbelbewegungen cntsprechen," Crcllc, 1858). Elliptic Integrals. 188. Attention has hitherto been restricted to integrations of rational algebraic functions, of logarithmic or circular functions, or of such functions as could be transformed to depend on these ; or, if irrationalities were introduced, they were such as involved the variable under the radical in no higher than the second degree. But the founders of the infinitesiiiial calculus early perceived that many integrals did not admit of expression by means of these ele mentary functions with which they were familiar. Apparently it was the geometrical interest attached to such integrations which first attracted notice. Thus James Bernoulli published, in the Ada, Eru- ditorum for 1691, a paper on the helicoidal parabola, in which we meet with the idea of comparing arcs of one and the same curve, which cannot be superposed. This spiral is the locus of the extremities of the ordinates of a parabola when its axis is rolled as a tangent to a fixed circle, the ordinates being measured towards the centre. The polar