## An Elementary Treatise on Algebra: To which are Added Exponential Equations and Logarithms |

### From inside the book

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Page 86

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**limit**their values ; such , for instance , as that they are to be whole numbers . But their investigation depends , in such cases , upon the particular properties of differ- ent numbers , and belongs , therefore , to the Theory of ... Page 222

... In this case , then , the number of permanences must by the preceding division of the proof be greater in the row which corresponds to the greater Sturm's Theorem . of the two

... In this case , then , the number of permanences must by the preceding division of the proof be greater in the row which corresponds to the greater Sturm's Theorem . of the two

**limits**p and q 222 [ CH . VIII . III . ALGEBRA . Page 223

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**limits**p and q , than in the row which corre- sponds to the less of these two**limits**, by a number which is exactly equal to the number of real roots contained between p and q . Fifthly . When the given equation has equal roots , u and ... Page 233

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**limit**of the number of positive roots is , therefore , by art . 294 , equal to the excess of the whole number of pairs of successive signs of the terms , over the number of per- manences ; that is , it is equal to the number of ... Page 234

... permanences are changed into variations ; so that ( m + 1 ) roots can be neither positive nor negative , and are , consequently , imaginary . Superior

... permanences are changed into variations ; so that ( m + 1 ) roots can be neither positive nor negative , and are , consequently , imaginary . Superior

**Limit**of Positive Roots . Secondly . If the 234 [ CH . VIII . § III . ALGEBRA .### Other editions - View all

### Common terms and phrases

126 become zero 3d root arithmetical progression coefficient commensurable roots common difference contained continued fraction continued product Corollary deficient terms denote derivative Divide dividend division equal roots equal to zero equation x² factor Find the 3d Find the 4th Find the continued Find the greatest Find the number Find the square Find the sum Free the equation Geometrical Progression given equation given number gives greatest common divisor Hence imaginary roots last term least common multiple letter logarithm monomials multiplied number of real number of terms polynomial positive roots preceding article Problem quantities in example quotient radical quantities ratio real roots reduced remainder required equation required number row of signs Scholium Second Degree Solution Solve the equation square root Sturm's Theorem subtracted Theorem unity unknown quan unknown quantity variable whence