OF THE ROOTS OF ALGEBRAIC EQUATIONS. 359 



24. The investigations of this paper have been restricted to ordinary algebraic equations, 

 nevertheless some of the results are of a more general character and need not be so restricted. The 

 proposition contained in Art. (8) is, I believe, perfectly general, as also is the proposition of 

 Art. (10) which is an extension of the former. In fact the theorems about maxima and 

 minima will be true for all such points as do not involve a failure of Taylor's Theorem, which 

 never occurs in the case of a rational algebraical function. The propositions concerning the number 

 and position of infinite branches are of course applicable only to algebraic equations. I will just 

 notice one instance of an equation not algebraic: suppose 



f(x) = smx=0 (32), 



"{•44^ 1 



(33), 



then z = ( cos v -— ] sin x 



\1 il 

 e> + e-'-' 



— sin X 



and 0= (sin ?/— j sin a' 



(35 J 



= cos * [e" - e~"\ (34). 



In equation (34-) the variables x and y are entirely separated; the factor e^ - e~^ when equated 

 to zero gives, as will easily be seen, only one real value of y, namely y = o ; this corresponds to the 

 real plane, and if we make ^ = in (33), that equation becomes 



X = sin X, 



and we have the ordinary figure of sines in the real plane. 



If we consider the factor cos.r in (34), we have an infinite number of real roots for the equation. 



namely .r = ± — , ± — , ± — , &c., and substituting these in (3P>), that equation becomes 



e" + e-" 

 z = ^ , 



which shews that from the maximum and minimum points of the real branch of the curve imaginary 

 branches set off in planes at right angles to the real plane, which are in fact common catenaries, the 

 directrices of which are in the plane of xy, and which go off alternately to positive and negative 

 infinity. 



2.5. In concluding this paper I will observe that I am not sufficiently well acf[uainted with the 

 literature of the subject to he certain as to how far the idea of it lias been anticipated. I will 

 observe, however, that in the late Mr. Murphy's Treatise on the Theory of Equations, (published 

 under the direction of tlie Society for the Diffusion of Useful Knowledge,) the existence of the roots 

 of Algebraic Ecjuations is demonstrated upon principles similar to those which I have adopted ; it 



