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. 1867 Excerpt: ... g(c), jti(c) of c, we should find that the given equations =0, Fi(x, y, z, c, qp(c), (c) =0, with their partial derived equations of the first order, making in all twelve equations, would involve twelve quantities to be eliminated; viz., dc dc ( 2c fi'c d-c ' dx' d'if' 3P hence the elimination cannot be effected, ...
Read More
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. 1867 Excerpt: ... g(c), jti(c) of c, we should find that the given equations =0, Fi(x, y, z, c, qp(c), (c) =0, with their partial derived equations of the first order, making in all twelve equations, would involve twelve quantities to be eliminated; viz., dc dc ( 2c fi'c d-c ' dx' d'if' 3P hence the elimination cannot be effected, except in special cases. Passing to the partial derived equations of the third order, wo should then have in all twenty equations, with, eighteen quantities to bo eliminated; viz., the twelve above given, and d3c dzc d3c d3c, ., .. dp' dxd additional: and we may therefore have for our results two partial differential equations of the third order between x, y, z; the latter being the dependent variable. In certain cases, it is unnecessary to make as many differentiations as have been indicated to enable us to effect the desired eliminations. Suppose, for example, "that the given equations contain but three arbitrary functions, p(c), qi(c), qp, (c): in this case, m = 3, 2m--1 = 5; and, to effect the eliminations, it would be generally necessary to form the derived equations of the fifth order, and we should have for our results three partial differential equations of the fifth order between o;, y, z. But if the arbitrary functions are so related that qi(c) = qp'(c), .gi2(c)--p"(c), the proposed equations be come Fx, y, z, c, (f(c), qr'(c), g"(c) = 0, Fl x, y, z, c, p(c), g-'(c), p"(c) = 0; and these, with their derived equations of the first and second orders, make twelve equations, involving the eleven quan tities dc dc d'-c dc dc C' fix' cly' &" 'dxdy' dy' 9(0, T'(c), qp"(c), jp'"(c), p""(c); and the elimination will lead to a single partial differential eqyation of the second order....
Read Less
