# Advanced Calculus, Revised Edition by Lynn H. Loomis, Shlomo Sternberg

By Lynn H. Loomis, Shlomo Sternberg

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These examples lead us naturally to the concept of the envelope of a family of curves. Let the family of curves y(x,y,C) = 0, (78) be given, where G is an arbitrary constant. e. the tangent at any given point of the envelope is also a tangent to the curve of family (78) that passes through this point, We derive the rule for finding the envelope. We start by finding the slope of the tangent to a curve of family (78). We differentiate equation (78), whilst taking into account t h a t y is a function of x and C is a constant; this gives us dtp (x, y, G) .

V 1 ~-jr. which gives, on substituting from equation (24): Mix) V"=-±L- (25) We now suppose t h a t the only concentrated forces are at the ends of the beam, being equal respectively to P0 and Pt (in the case of Fig. 19, P0 is nega tive) ; in addition to these, there are bending couples at the ends, the moments of which will be denoted by M0 and M^ The distributed loading per unit length of the beam is denoted by f(x). We find the sum of the moments of the external forces acting on the part NL of the beam (Fig.

Integration again gives a family of rectangular hyperbolas, referred in this case to the axes of symmetry: x2 — y* = 0. As may easily be seen, this family is obtained from the given family (94) by turning it through 45° about the origin. e. the orthogonal trajectories of family (92) consist, for m > 0 , of a family of similar ellipses, and for m < 0, of a family of similar hyperbolas. The orthogonal trajectories of the parabolas y = Cx2 are illustrated in Fig. 14. § 2. Differential equations of higher orders; systems of equations 13.