their greatest generality let us suppose all the variables and differentials to receive variations. Let When a involves more than three independent variables, the expression for the variation of / 2 is of course similar. 307.] Suppose however that an equation of relation is given between the variables and their differentials which are involved in n; and, to fix our thoughts, let us take the case of three variables X, y, ; and suppose the equation to be l=f{x,dx, d’x, ... y, dy, dạy, ... 2, dz, d’z, ...} = 0. (63) If l involves only x, y, z, z may be expressed in terms of x and y, and thence dz, d’z, ... may be found, and substituted in , so that a will become a function of only two variables, x and y: but as l involves the differentials of the variables, such an elimination is generally impossible, and we are obliged to have recourse to the following process. Take the variation of l, viz. (61), a result similar to (62) will be deduced from (67), wherein instead of x will be x + 1€, instead of x,,X, +1&v ... instead of y, Y+in, ..., and so on for the others; and as a is undetermined, we may consider the variation to involve three independent quantities. The variation will also be found in a similar manner if the original element-function involves more variables, and if these are related to each other by many equations of condition. 308.] Suppose however that the element involves three variables x, y, z; that x is equicrescent, and that y and z are two unknown functions of x, and independent of each other, and that the quantity whose variation is to be calculated is ri dy dạy dmy dz daz d"z) wherev=P(x, y, da dr ,... dam, z, Ta' d. 22).. dxn), f being a known function. To give v the most general variation, let us suppose that not only x, y, z, but that also the derived functions of y and z vary: then, adopting the following substitutions, da y el 9 = y(); daca +32 - dz + -...(-)"0."}«, dx ; (69) an expression consisting of two parts; of which one involves the values of the variables and their variations at the limits only; and the other involves a process of integration, and which cannot be performed unless the relations between x and y and z are given. These several parts admit of explanation similar to that which has been given of (56) in Art. 304. Also it admits of a geometrical explanation similar to that of Art. 304; the relations between x and y and between x and 2 represent two cylinders which are perpendicular to the planes of (x, y), and of (x,z) respectively; and these by their intersection define a curve in space. Let us consider the general displacement of a point on this curve to be due, (1) to two displacements perpendicular to each other in the normal plane, and (2) to one along the tangent line; now by a process exactly analogous to that of Art. 304 it may be shewn that the quantities under the signs of integration involve the normal displacements only; and that vox is the only term wherein the tangential displacement appears. If v contains any number of undetermined functions, the variation of v dx will be 'calculated in a similar manner, and will consist of a series of terms and quantities similar to those of equation (69). 309.] In the last article y and are considered to be independent of each other; if a relation is given connecting them and their derived-functions and x, and of the form multiplying which by an undetermined constant à, and adding it to dv, we have comparing which expression with that of (68), and noticing the process by which (69) is deduced from (68), it is palpable that (72) will lead to a result of the form given in (69), and with quantities such that in the place of y will be y+ 1); in the place of Y', Y+ (),...; in the place of 2, 2+1 (45); in the place of 2', z' + c ),...; and so on : and thus the variation will be reduced to the form of a definite integral, whose element-function involves x and two unknown and independent functions of x. 310.] Certain processes in the sequel will require the calculation of the variation of a variation, that is, of the second variation of a definite integral. As the principles and the method are the same as those explained and applied in the preceding articles, I |