tween their variations with which the former part of (51) must consist. If x is assumed to be equicrescent, so that dex = d3x... dx = 0; = 303.] If however the element of the definite integral is expressed in terms of differential coefficients, in which ≈ is equicrescent, so that where u = v dx, dy day dry), F representing a known function, and the relation between x and y being undetermined; then to give to v the most general variation dy day that is possible, x, y, dx' dx2 will all vary. For convenience of notation let us substitute as follows; so that dv v V = F(x, y, y', y",... y(")); therefore (dr) &r + (dv) by + (dv) by ' + ... dy dv dv dv = (dv)dx+ (dy) dy + (dy) dy + which expression*, it will be observed, consists of two parts; the former depends on the values of the variables at the limits of integration; the latter involves an integration, which cannot generally be effected unless the function connecting x and y is known. [ 0 As to the former part, the first term is vdx where v1 and v1 are the values taken by v at the limits; these do not involve dy, but only involve de at the limits; and are the variations of the definite integral due to the change of the limits, being the quantities which have been determined in (86) and (87), Art. 96. The other terms contain ∞ and its derived functions; and if dx = 0, w = dy, all these terms involve the values which ty and its derived functions take at the limits, and are independent of the general functional connection between x and y. 301.] Certain other properties of (56) require notice. If w=0, that is if dy-y'dx=0, dy = ; * In the Memoir on this Calculus by M. Poisson, which was read to the French Academy in 1831, and is printed in Vol. XII of the Memoirs of the Institute, equation (56) is deduced from first principles. PRICE, VOL. II. 3 K and thus the ratio of the variations and of the differentials of y and a is the same: and in this case, 1 8. ['v dx = [v dx]' = v, dr, —v, &r。 i (57) that is, if we make the coordinates of a point on a curve to vary, so that the ratio of the variations and of the differentials of the coordinates is the same, we do not leave the curve, but pass to a consecutive point of it, and the definite integral is increased by the value of its element-function corresponding to the superior limit, and diminished by that corresponding to the inferior limit. Also the geometrical meaning of o da deserves notice. Let the variations of the coordinates of any point on the plane curve under consideration be dr and dy, and let the projections of the space through which the point (x, y) has moved be estimated along the tangent and normal of the original curve at the given point, and let these projections be r and v; then substituting which values in (56), it will be seen that every term in the part at the limits except the first involves only v, the VT ds. normal displacement, and the part of that involving 7 is [v, da ]', [vox], if the variation is made on the which is equivalent to v dx supposition that v = 0. Also the part under the sign of integration involves v only; the reason of which is, the variation in the form of a curve due to the shifting of its several points and elements arises from the infinitesimal normal displacement only; the effect of the tangential displacement being to shift a point to another consecutive point on the curve. 305.] In reference also to the general expression (51) it is worth remarking, that if dr and dy are replaced by dx and dy, that is, if the shifting of the point takes place along the curve only, and if there is no normal displacement, then the total 1 ♫ is that which takes place at the limits; thus in and the last two terms, after integration by parts, become 1 + [dx, dx+ dv, dy]' — [x, d2x+v_d°y] + [ '{x,dx+\,d°y} 2 and so on. Hence 1 0 ô. [ 'î = [ '{x dx + x1 d.dæ + x» d.d2x + .....+x„d.d”x Ω X1 ; by reason of equation (48); so that the total variation is reduced to the difference between the values of the infinitesimal element at the first and the last limits. 306.] We proceed now to investigate the variation of a definite integral whose element involves three independent variables and their successive differentials; and to consider the variations in |