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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. 1830 edition. Excerpt:...proposed cannot be rendered an exact fluxion, whatever be the form of t. 7. We should now proceed to find a factor which renders the formula integrable per se. The process is the same as that adopted, ch. 4, art. 28; but the resulting equations, as in that article, are as difficult to integrate as the proposed formula. They will contain not only the variables and their fluxional coefficients, but also t and its fluxional coefficients-r-, We shall not stop to consider the dx dy dz r cases in which t can be found. If the proposed be homogeneous, it may be shown as in ch. 4, 29, case 2, that 1 t =;. px + ay + s.z 8. Required to integrate a formula of three variables which is independently integrable. The rule is the same as that of ch. 4, 24, rule 2, where there are only two variables: 'First integrate on the supposition that one of the variables as z is constant, in which case the correction is z a sole function of z; then differentiate the result partially with respect to s, by which the value of z is found and the required fluent obtained.' It may be sometimes convenient to begin by integrating on the supposition that two of the variables are constant; in which case the correction is an implicit function of those variables; and we must then differentiate the result with respect to both of them. It appears from art. 3 that this is not an exact fluxion; but that its two first members form an exact fluxion, whose fluent is x + y +--. It also appears from equation (/3) that it is independently integrable; wherefore xz du x dz x + y u = x.+ y H 1-z = 0.-.-7-= (--j-=-, or y dz y dz z dz x+y x z dz dz---j-= 1 =--=--. or z = cz, and the dz z y z z z required primitive is xy + xz + y = cyz. Ex. 6. (y% + yz + z)dx + (x + xz + z)dy + (xi + xy + y)dz = 0. dx...