Rev. R. Murphy on the Syminetrical Functions of Roots. 413 



coexistence of unequal elasticities is not only possible, but must 

 be the case ; for if the aether be an elastic fluid uniformly dif- 

 fused through space absolutely devoid of other matter, it fol- 

 lows that wherever it penetrates a space filled with any me- 

 dium, such as air, it must necessarily be attracted round each 

 particle of air, and form spheres of a density increasing towards 

 their ceiitres. Amongst an infinite number of such spheres 

 uniformly diffused, a succession of vibrations communicated in 

 a given direction, will of course give rise to vibrations propa- 

 gated with various velocities, according to the particular elasti- 

 city of different parts of the disseminated medium : thus we 

 shall have a number of coexistent vibrations producing un- 

 dulations of different lengths which, when they are incident 

 upon a new medium, will cause a deviation in position pro- 

 portional to the unequal lengths of their undulations, accord- 

 ing to the well known and established explanation of the 

 general law of simple refraction as expressed by the undula- 

 tory theory. 



November 1st, 1831. 



LIII. On the Symmetrical Fiinclions of a specified Number of 

 the Roots of an EqiiatioTi. By the Rev. R. Mu rphy. Fellow 

 of Cuius Coll. and of the Camb. Phil. Soc. 



To the Editors of the Philosophical Magazine and Annals. 

 Gentlemen, 



IN a paper published in the last Number of the Transactions 

 of the Cambridge Philosophical Society, I have given a 

 few simple rules relative to the solution of algebraical equa- 

 tions, with their demonstrations. The sum of any specified 

 number of the roots taken in order from the least, upwards, 

 and the sum of any given function of such roots, may be 

 thence found, for any proposed equation (p{x) = 0, containing 

 only positive and integer powers of x. The coefficients of the 

 difierent terms of an equation are, as is well known, the sums 

 of symmetrical functions of all the roots ; and my present 

 object is to show a simple method of obtaining the corre- 

 sponding sums of the symmetrical functions of a specified num- 

 ber oi the roots; and as general investigations relative to a 

 specified number of the roots are I believe new in analysis, 

 this paper may not be unacceptable to your mathematical 

 readers. 



Let«, , a^, «;, a„ be the m least roots, taken in order, 



of the e(|uation '<^ {x) = 0, and A any arbitrary quantity. Then 

 by a rule given in (§ 3) of the paper above referred to, we get 



log 



