On the Integration of Linear Differential Equations. 187 



view; for however indispensable these minute molecular changes 

 may be for the production of thermo-electrical currents, they do 

 not appear sufficient to account for the chemical effects obtained ; 

 but if it could be shown that the contact of dissimilar metals, or 

 of dissimilar particles of the same metal, while they undergo a 

 slow change by the action of heat, can so react on the heat as to 

 derive electricity from it, then the apparently wide difference be- 

 tween the chemical effect derived from a thermo-electrical couple 

 and the minute action on the joint of it may be more easily 

 comprehended. 



Bismuth, in addition to its being by far the best metal for 

 supplying the place of a positive thermo-electric bar, has been 

 shown in the valuable magnetic researches of Dr. Faraday, to be 

 the most energetic diamagnetic substance at present known. It 

 is likewise remarkable in its alloys. In the gold coinage a mi- 

 nute addition of bismuth greatly impairs the durability of the 

 coin ; and with lead or tin, mixtures are made which melt at 

 lower temperatures than the melting heat of the metals combined 

 together. These properties are due to conditions of the forces of 

 aggregation of which science is at present able to give no ac- 

 count, but which appear to point to an ample field for inquiry 

 for the future, in which the properties of thermo-electricity, 

 when further unmasked, may render much assistance. 

 I remain, Gentlemen, 



Yours very respectfully, 



Richard Adie. 



XXIX. On the Integration of Linear Differential Equations. 

 By the Rev. Brice Bronwin*. 



THE present paper is an addition to one published in the 

 Number of this Journal for December last, and is intended 

 to illustrate the use of the arbitrary function \(D). 



In the Cambridge Mathematical Journal, first series, vol. ii. 

 page 193, is given a solution of 



xD m u —pmD m ~ 1 n + Lvu = 0, p integer. 

 By the conversion of symbols this may be solved as an equa- 

 tion of the first order, but it serves to show the use wbich may 

 be made of the : ymbolical function X(D). Make 



tlicn 



vru=~Dxu + \(D)u, and xu = ~D~ 1 vju — D _1 \(D)h. 

 In the last, change u into D'"//, and we have 



xD m u = D-'mD m u-ir , \(\))D m u. 

 * Communicated by the Author. 



