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We obtain an equation of the form f(x, a, ft) = 0, (3) giving the abscissa of the point of contact of the curve with its envelope. Since the curve touches its envelope, equation (3) must also be true for a contiguous value of x, viz., x + Sx (unless the tangent at the point of contact be parallel to the axis of y, in which case we could have eliminated x between (1) and (2) and proceeded in the same way with y). Hence f(x, a, b) = 0, (4) f(x+Sx, a, b) = 0.1 (5) The latter may be expanded in powers of Sx, when it becomes, df.' f(x, a, b)+dJxSx+. = 0, (6) and therefore in the limit I= If, then, x be eliminated between f(x, a, \/3) = 0, f(x, a, \/3) = 0, we obtain the relation sought. It will be observed that this is precisely the same process as finding the envelope of jx, y, a, \/3)=0, considering a, \/3 as the current co-ordinates and x, y as parameters connected by the relation F(x, y) = 0. Ex. Given that x+y=$ is the envelope of-+Y=lt find the necessary relation between a and b. 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Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1886 Excerpt: .viz., F(x, y) = 0 (2) Required the relation between a and ft. Eliminate y between (1) and (2). We obtain an equation of the form f(x, a, ft) = 0, (3) giving the abscissa of the point of contact of the curve with its envelope. Since the curve touches its envelope, equation (3) must also be true for a contiguous value of x, viz., x + Sx (unless the tangent at the point of contact be parallel to the axis of y, in which case we could have eliminated x between (1) and (2) and proceeded in the same way with y). Hence f(x, a, b) = 0, (4) f(x+Sx, a, b) = 0.1 (5) The latter may be expanded in powers of Sx, when it becomes, df.' f(x, a, b)+dJxSx+. = 0, (6) and therefore in the limit I= If, then, x be eliminated between f(x, a, \/3) = 0, f(x, a, \/3) = 0, we obtain the relation sought. 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Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1886 Excerpt: .viz., F(x, y) = 0 (2) Required the relation between a and ft. Eliminate y between (1) and (2). We obtain an equation of the form f(x, a, ft) = 0, (3) giving the abscissa of the point of contact of the curve with its envelope. Since the curve touches its envelope, equation (3) must also be true for a contiguous value of x, viz., x + Sx (unless the tangent at the point of contact be parallel to the axis of y, in which case we could have eliminated x between (1) and (2) and proceeded in the same way with y). Hence f(x, a, b) = 0, (4) f(x+Sx, a, b) = 0.1 (5) The latter may be expanded in powers of Sx, when it becomes, df.' f(x, a, b)+dJxSx+. = 0, (6) and therefore in the limit I= If, then, x be eliminated between f(x, a, \/3) = 0, f(x, a, \/3) = 0, we obtain the relation sought. 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