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XI. Analysis of the Roots of Equations. By the Rev. R. Murphy, M.A., Fellow of 
Caius College, Honorary Member of various Philosophical Societies. Communi- 
cated by J. W. Lubbock, Esq. F.R.S. 
Received April 6, — Read April 27, 1837. 
i. The object of this memoir is to show how the constituent parts of the roots of 
algebraical equations may be determined, by considering- the conditions under which 
they vanish, and conversely to show the signification of each such constituent part. 
2. In equations of degrees higher than the second the same constituent part of the 
root is found in several places governed by the same radical sign, but affected with 
the different corresponding roots of unity as multipliers. 
3. The root of every equation, of which the coefficients are rational, contains a ra- 
tional part, for the sum of the roots could not otherwise be rational. 
This rational part, as such, is insusceptible of change in the different roots of the 
same equation, consequently its value is the coefficient of the second term (with a 
changed sign) divided by the number of roots, or index of the first term. 
4. The supposed evanescence of any of the other constituent parts implies that a 
relation exists between the roots ; if such a relation be expressed by equating a func- 
tion of the roots to zero, that constituent part will be the product of all such func- 
tions, and a numerical factor. 
5. The joint evanescence of various constituent parts implies the co-existence of 
various relations between the roots, and thus an interpretation may be given to each 
of the constituent parts, riveting the expression of the root in the memory, and beau- 
tifully converting the solution of a problem into a condensed enunciation of various 
theorems. 
For simplicity these principles are first applied to equations of lower degrees. 
6. Let us take for example the general quadratic equation 
x 2 + a x -J- b, 
the two roots of which are represented by x l9 x 2 in the formulae, 
x \ — % ~h s/ & 
0, a relation is then established between the quantities 
X j X 2 — 0, 
Y 
To find a suppose a = 
and <r 2 , viz. 
MDCCCXXXVII. 
