Archive image from page 49 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni19rees Year: 1819 INDETERMINATE ANALYSIS, may be obtalncc?. But as thefe queftions generally admit of a great number of folutions, the objcft of enquiry is not fo much to find the values of the intermediate quan- liliev as to determine <5/rn'ori the number of them that the equatioa admits of; and this, therefore, ftiall form the fub- jecl of our future inveftigation. Now we have feen, that in the equation ax -\- by - - c, the n
Archive image from page 49 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni19rees Year: 1819 INDETERMINATE ANALYSIS, may be obtalncc?. But as thefe queftions generally admit of a great number of folutions, the objcft of enquiry is not fo much to find the values of the intermediate quan- liliev as to determine <5/rn'ori the number of them that the equatioa admits of; and this, therefore, ftiall form the fub- jecl of our future inveftigation. Now we have feen, that in the equation ax -\- by - - c, the number of folutions is generally exprefled by cp eq and q being firft determined by the equation ap â b q z=. I. If, therefore, in the equation a X -\- b y â =. d - c Zf â¢we make fuccefTively z = i, 2, 3, 4, &c. the number of fo- lutions for each value of will be as below ; viz. a X -{⢠o y = a â c, number or lolutions- + hy = d ic ax + by â J â 3r &c. &c. -the fum of which will be the total number that the given equation admits of; and therefore, in order to find the exad number of folutions in any equation of this kind, we muft firft afcertain the fum of all the integral parts of the arithmetical feries (id-c)p_(J~2c)p{d-3f)Pi±:if)l+Scc; and b b b b 1 {d-c a a a a and number of the difference of tie two will be the exaft intrcgal folutions. Now in both thefe feries, we know the firft and laft term, and the number of terms ; for the general term being V ( - g )P â_j '' -CT.)q .â_ ana > o a we fiiall have the extreme terms by taking the extreme li- mits of z, that is z = i, and x < â; which laft value of c alfo expreflcs the number of terms in the feries. Hence then, having tlie elements of the progrefiions given, the fum of the whole feries is readily obtained ; and if therefore we alfo find the fum of the fraftional parts in each, we fiwU have, by dedufting it from the whole fums, that of the integral part of the feries as rcquirca. The latter p
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