...divided by 1, «г, т", &с. the roots will all be divided by m. To transform the equation ф (.«) = 0 into one whose roots are the reciprocals of the roots of the former : assume л? = - , У we obtain aaaa ( W. 280— 97; Bour. 275— 80 ; G. 310— 25.) To take...

...2, 4, 8. = a .'. a = 4 EQUATIONS, WHOSE ROOTS ARE IN HARMONICA!. PROGRESSION. Transform the equation into one, whose roots are the reciprocals of the roots of the given équation, and are, therefore, in arithmetical progression. These latter roots may be found as in (3ic),...

...the proposed equation be the same when taken in an inverted order, it is evident that the equation, whose roots are the reciprocals of the roots of the given equation, will be identical with it, and will therefore furnish the same series of roots. The roots of the original...

...between 0 and -. 6 287. To find an inferior limit of the positive roots, we must transform the equation into one whose roots are the reciprocals of the roots of the former; and the reciprocal of tho superior limit of the roots of the transformed equation, found by...

...the proposed equation be the same when taken in an inverted order, it is evident that the equation, whose roots are the reciprocals of the roots of the given equation, will be identical with it, and will therefore furnish the same series of roots. The roots of the original...

...for x, we shall have if or, clearing of fractions, and reversing the order of the terms, an equation, whose roots are the reciprocals of the roots of the given equation. Hence, To transform an equation into another, whose roots shall be the reciprocals of the roots of...

...positive roots of the equation /(a;) = 0. any of the positive roots. For transform the proposed equation into one whose roots are the reciprocals of the roots of the proposed equation, and then the reciprocal of the superior limit of the positive roots of the transformed...

...is a number which is not greater than any of the positive roots. For transform the proposed equation into one whose roots are the reciprocals of the roots of the proposed equation, and then the reciprocal of the superior limit of the positive roots of the transformed...

...Remove the fractional coefficients from the equation 32. Reciprocal Roots.-—To transform an equation into one whose roots are the reciprocals of the roots of the proposed equation, we change x into - in the identity of Art. 30. This gives, after certain easy reductions,...

...is a number which is not greater than any of the positive roots. For transform the proposed equation into one whose roots are the reciprocals of the roots of the proposed equation, and then the reciprocal of the superior limit of the positive roots of the transformed...