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Published online by Cambridge University Press: 03 November 2016
There are many methods of solving algebraic equations whose coefficients are real and have given numerical values. In practice it very often happens that the coefficients are not given numerically, but are expressed in terms of certain parameters, and that what is wanted is not the roots of the equation for special values of the parameters, but an indication of the way in which the roots behave as these parameters vary between certain limits. For example, if the equation is the period equation of a mechanical or electrical system, we may wish to know how its roots vary as certain components of the system are changed, not merely the values of the roots for definite values of the components. The method sketched below has been found very useful for studying cubics from this point of view and applies quite well to quartics. It also provides interesting elementary examples in the theory of equations.
* This restriction keeps the real parts of the roots negative, and this is the case of practical importance, but the method applies to other values of k and c. The equation is the period equation of a feedback servomechanism; c depends on the circuit components, and k on the amount of feedback.
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