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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1868 Excerpt: ...the parameters are supposed to vary by infinitely small increments, the surfaces are said to be consecutive. Thus let F(x, y,, a) = 0, .... (1), be the equation of a surface, and let the parameter a, take an increment A, converting (1), into F(x, y, z, a + h) = 0, .... (2); then if h be supposed indefinitely small, the surfaces (1) and (2) will be consecutive. Moreover, the surfaces (1) and (2) will usually intersect, and their intersection will vary with the value.of h, becoming fixed and determinate when the surfaces are consecutive. 195. Prop. To determine the equations of the intersection of consecutive surfaces. To effect this object, we must combine the equations Fx, y, z, a) = 0, (1), and F(x, y, z9a + h) = 0, (2), and then make h = 0. By reasoning precisely as in the case of consecutive curves, (Art. 143) we prove that the two conditions Fx, y, z, a) = Q, .... and ff a) = 0, .... (3), must be satisfied at the same time. By combining these equations, so as to eliminate first y, and then x, we shall have the equations of the projections of the required intersection on xz, and yz. 196. Prop. The surface which is the locus of all the intersections of a series of consecutive surfaces, touches each surface in the series. If we eliminate the parameter a between the two equations F(x, !, z, a) = 0, ....(l), and dI Il = 0, .... (2), the resulting equation will be a relation between the general co-ordinates x, y, z, of the points of the various intersections, independent of the particular curve whose parameter is a, or in other words, the equation of the locus. Eesolving (2) with respect to a, the result may be written a = cp(x, y, z), and this substituted in (1) gives Fx, y, z, cp(x, y, z) =05.... (3), which will be the equation of the locus. Now dif
