. Differential and integral calculus. erential Calculus 125. To find the successive total differentials of a function of twoindependent variables. Let u = f(x,y); then, (§121), _ du du du = — doc -+- — dy ^•rr • • . . 8U - Sit Differentiating, remembering that — and — are, in general, functions of x and y, and that x and y, being independent, may-be regarded as equicrescent, we have, V2 &U J 2 J. &U d2U .^ ., dx2 dxdy dydx df d2u d2u d2u Similarly, we find, d^u dsu d^u d^u d*u = _ &3 + 3 __dx,dy + 3_^dxdf + _ dA and so on. By observing the analogy between the exponents of du, d2u, d*u, .

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. Differential and integral calculus. erential Calculus 125. To find the successive total differentials of a function of twoindependent variables. Let u = f(x,y); then, (§121), _ du du du = — doc -+- — dy ^•rr • • . . 8U - Sit Differentiating, remembering that — and — are, in general, functions of x and y, and that x and y, being independent, may-be regarded as equicrescent, we have, V2 &U J 2 J. &U d2U .^ ., dx2 dxdy dydx df d2u d2u d2u Similarly, we find, d^u dsu d^u d^u d*u = _ &3 + 3 __dx,dy + 3_^dxdf + _ dA and so on. By observing the analogy between the exponents of du, d2u, d*u, ...and those of the development of (x 4- a), (x + a)2, (x+a)s, . . we are enabled to write the value of dnu. The student may apply this process to any example. Direction of Curvature Points of Inflexion 173 CHAPTER XII. DIRECTION OF POINTS OF CARTESIAN CURVES. 126. A curve is concave upward or convex upward at a pointaccording as the tangent at the point lies below or above

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