. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature. Encyclopedias and dictionaries. INDETERMINATE a;NALYSIS. the folution of the equation x^ â y'' = C 2' is always to be ubtained; and in the following- propofition, with which we ihall conclude this article, it will be iQitw that every equation of the form x^ â Ay' = B z% which is poflible, may be transformed to another of the form x' â y'' =z cz''; and that *â¢, y, and z, in tlie original equation, will be dependent upon thofe of x', y', and z', in the transform/jd equation ; and tlierefore thele laft being known, th
. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature. Encyclopedias and dictionaries. INDETERMINATE a;NALYSIS. the folution of the equation x^ â y'' = C 2' is always to be ubtained; and in the following- propofition, with which we ihall conclude this article, it will be iQitw that every equation of the form x^ â Ay' = B z% which is poflible, may be transformed to another of the form x' â y'' =z cz''; and that *â¢, y, and z, in tlie original equation, will be dependent upon thofe of x', y', and z', in the transform/jd equation ; and tlierefore thele laft being known, the former will be known alfo. PUOP. XIV. To transform evei-y poflible equation of the form *â '' â Ajr^ = B, to another dependent equation â of the form x"- ây"' â c z'\ â £x. I.âIt is required to transform the equation x" â Sy^= 11 z", to another of the form x"' â y''^ := c z": Having firft afcertained the pofTibility of the equation by the foregoing propofition, the transformation may be effect- ed in the following manner. Affume X = ny ~ 1 iy\ and this fubflituted for x, gives Take n, fo that n" â ^ may be divilible by 11 ; that is, let n = 4, and our equation becomes ;^_y' + IIy"- = z^; or {y -\y'Y - 5 y"' = ~ or, by making y â ^y' = x, we have â¢*'"" 5y"~ ^' i "'â ' x"â z" = S y", as required, that is, the equation has been reduced from the form x" â 5y'= II 2% to another of the form a'" â y" = cz'\ or, at leaft, to x'"- â z'^ = ^ y'-; which differs from the forego- ing only in the letters. And, by means of the values of .v', y', and 2, in this laft, we readily arrive at thole of *â¢, y^ and z, in the one propo- rtion ; for x'= y â 4^', or y = x' + /\.y' and X ^= nyâ 11 y', or x = 4j» â 11 j' Now we have feen, under tftn article Diopiiantike, that the general values of ^'.and z, in equations, of the form x': â 2^= 5_j''% are ^'= / +
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Keywords: ., bookcentury1800, bookdecade1810, booksubjectencyclo, bookyear1819
