will consider only one simple instance: viz. that in which it is 1 required to find 82u, when u = =['v dx, and dv dv but 8v = (dv) &x + (dy) by + (dry) by " + (dy )ży" +. 82v = ô (dv) 8x + 8 8 (dv) by + 8 (dv) by2 + dy dy dv + (dv) 32x + (dv) 83y + (dy ) 8y + dx dy (dv)... dy' 8x (73) are functions of x, y, y', y”.....; 8. (dr) = (day) &x + ( (day) by + (dvd) by + + 2 { (dry) 8x by + ( 127 ) 8.x by + ... + (d) by by +... }; (75) 2{( d2v dx and therefore dx dy' By similar processes may d3u, d'u,... be calculated. 311.] We come now to the investigation of the variation of a multiple integral; but instead of taking the problem in its most general form, I will consider the case of a double integral only; for the principle on which the inquiry is founded is the same in all cases; and the number of terms in the result increases so rapidly with each new integral sign, that by taking any higher order the formulæ are so complicated as to require new symbols and new modes of abbreviation, and no result useful for our present purpose is arrived at. And I shall consider only a simple case of a double integral: that, viz. in which the element-function involves x, y, z, (z being an undetermined function of x and y), and the partial derived functions of z of the first and second orders; and in which also the limits of integration are given by an inequality in accordance with the principles explained in Art. 214: so that the range of integration includes the values of the variables corresponding to all points within a given closed surface. This case will suffice for all the examples to which I shall at present have occasion to apply the calculus; and the student who desires further information will find an investigation of the general case in those memoirs of Ostrogradsky, Sarrus, Delaunay, and Lindelöf, which are mentioned in the foot-note of page 411. In those by Sarrus and Lindelöf especially the difficulties of the variations of double definite integrals are elucidated, and to them I am under great obligation for the following investigation. A peculiar symbol, which they call the symbol of substitution, has been largely employed by them; I however have found the symbols already employed in this work sufficient for the purpose. Let the double definite integral which is the subject of vari and let us suppose v to be a function of x,y,z, and of d2 z dx2 x and y. d2z dx dy (77) dz dy -), (d), where z is an undetermined function of dy2 For the sake of abbreviation let the following symbols du= the upper and lower accents referring to the x- and y- partial Υ substituting for 8.dy dx from equation (23). And integrating Χρ Yo = Yo Χρ XoYo Now let us assume that in the general variation of the ele- where w', w,, w", ... are partial derived functions of w. (83) Let z, z, z,, z",... denote the partial differential coefficients of v with respect to z, z, z,, z",... respectively; then so that du = dv = zdz +z′dz′+z,dz,+z′′dz′′ + z ̧ ́dz, +z,„„dz,„, Χρ + Το ΤΟ Χρ (84) (z w + z 'w' + z,w,+z′′ w′′ +z;w; +z,w)dy dx. (85) Now the last part of this equivalent of du involves not only w, which is an arbitrary function of x and y, but also its x- and y-partial derived functions of the first and second orders; and as it is to be integrated with respect to the same variable for which it is differentiated, several of these terms are capable of considerable simplification by means of integration by parts, and of the theorems for the variation of definite double integrals which are given in Art. 217. Thus taking the x-integral of this equivalence with the limits Again, taking the last three terms in the latter part of the second member of (85), as they involve second derived functions of w, two integrations will be required for their simplification. Let us first take the term whose element-function is z"w"; then, by the theorem expressed by (87), The first term of this second member does not admit of further reduction, because the element-function of a y-integral involves two x-derived functions. The second and third terms admit of the following simplification. |