An Elementary Treatise on Algebra: To which are Added Exponential Equations and Logarithms |
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Page 220
... sign to U. 289. Definition . A pair of two`successive signs , in a row of signs , is called a permanence when the two signs are alike , and a variation when they are unlike . 290. Sturm's Theorem . Denote the first member of the ...
... sign to U. 289. Definition . A pair of two`successive signs , in a row of signs , is called a permanence when the two signs are alike , and a variation when they are unlike . 290. Sturm's Theorem . Denote the first member of the ...
Page 221
... row of signs corresponding to the values of for any value p 9 of the variable . u , U , U ' , U " , & c . , of the variable , and also for a value The difference between the number of permanences of the first row of signs , and that of ...
... row of signs corresponding to the values of for any value p 9 of the variable . u , U , U ' , U " , & c . , of the variable , and also for a value The difference between the number of permanences of the first row of signs , and that of ...
Page 222
... sign of a term , which is neither the first nor the last of the series , does not increase or diminish the number of permanences of the row of signs . Thirdly . When the first term u of the series , in chang- ing its sign , vanishes ...
... sign of a term , which is neither the first nor the last of the series , does not increase or diminish the number of permanences of the row of signs . Thirdly . When the first term u of the series , in chang- ing its sign , vanishes ...
Page 223
... row which corre- sponds to the less of these two limits , by a number which is exactly equal to the number of real ... signs as the given series or the reverse signs , so that each pair of successive signs is of the same name , whether ...
... row which corre- sponds to the less of these two limits , by a number which is exactly equal to the number of real ... signs as the given series or the reverse signs , so that each pair of successive signs is of the same name , whether ...
Page 224
... row of signs is + , + , + , + ; and when x = co , it is + , - , - , + ; there are then two real roots . Again when x2 , the row of signs is + , + , + , + ; and when x1 , it is + , - , - - ; + ; the two real roots are therefore both ...
... row of signs is + , + , + , + ; and when x = co , it is + , - , - , + ; there are then two real roots . Again when x2 , the row of signs is + , + , + , + ; and when x1 , it is + , - , - - ; + ; the two real roots are therefore both ...
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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