Archive image from page 56 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni19rees Year: 1819 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
Archive image from page 56 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni19rees Year: 1819 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 '= / + 59 .y' = 2pq â¢whence wc have, for the general val
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