264 Mr. B. A. Murray on a rigorous Demonstration 



moving from or towards any centre within the body, by spherical 

 dispersion or concentration, portions of matter lying in complete 

 and uniform spherical shells. 



In the above investigation I have supposed the imaginary 

 body to consist of materials (positive and negative) following one 

 continuous law. But I might have placed a symbol X before all 

 the formulas, and supposed the imaginary body to be made up 

 of the superposition of any number of bodies each following its 

 own law and having its own origin of coordinates. The result 

 is that the materials may be transferred, in the way described in 

 the last paragraph, not about a single centre only, but about any 

 number of centres in the body. 



J. H, Pratt. 



XXXV. A rigorous Demonstration, by elementary Geometry, of 

 the Proposition usually classed as the Twelfth Axiom of Euclid. 

 By B. A. Murray, Esq. 



To the Editors of the Philosophical Magazine and Journal. 



2 Palace Street, Dublin, 

 Gentlemen, March 9, 1867. 



I BEG the favour of an early insertion in your Magazine of 

 the accompanying paper, and trust it will prove not un- 

 suitable to your valuable pages. 



It will be found that I have not in the course of demonstration 

 expressly adopted or tacitly assumed any new postulate ; and the 

 corollary to the second principal proposition being proved, the 

 methods of carrying out the demonstration, I need scarcely say, 

 are unnumbered ; mine will, I hope, shortly appear in a little 

 brochure nearly ready for publication. I have not deemed it ne- 

 cessary for your class of readers to refer to the postulate, axiom, 

 or previous proposition obviously authorizing the ordinary steps 

 of the demonstration. They will appear elsewhere in the proper 

 school form. 



I am, Gentlemen, 



Most truly yours, 



B. A. Murray. 



Preliminary. 



(a) Two quadrangles are equal, if they have each three sides 

 and the two angles comprised by them respectively equal, 



(b) In a quadrangle, if two opposite sides and two adjacent 



