378 Mr. A. Cayley on a Result of Elimination. 



which correspond to acetic and benzoic sethers. Thus, 

 C«H^I + AgO^C^H^O^ = Agl 



Iodide of propylenyle. Acetate of silver. Iodide of silver. 

 + C6H5 0, CH^QS, and C^H^I 



Acetate of propylenyle. Iodide of propylenyle. 



AgO, C'-* H^ 03 = Agl + C^H^O, C'4H*03. 

 Benzoate of silver. Iodide of silver. Benzoate of propylenyle. 

 Zinin has also obtained a carbonate of propylenyle correspond- 

 ing to carbonic sether. When mercury is acted upon by iodide 

 of propylenyle, a metallo-organic compound is formed, which cor- 

 responds to that obtained by Frankland as the result of a similar 

 reaction with iodide of aethyle. 



XLVII. Note upon a Result of Elimination. 

 By A. Cayley, Esq.* 



TF the quadratic function 



{a,b,c,f,g,h\oc,y,zy^ 



break up into factors, then representing one of these factors by 

 ^x + 'qy-^-^z, and taking any arbitrary quantities a, /3, 7, the 

 factor in question, and therefore the quadratic function is reduced 

 to zero by substituting /S^— a??, y^— af, «?;— /3f in the place of 

 X, y, z. Write 



{a,b,c,fg,Iifpi^-a.r], y^-cc^, «77-^f)'= (a,b,c,f,g,h][a,/3,7)2j 

 the coefficients of the function on the right hand are 

 a= . C7?2 +6^ -2fv^ . 

 b= cf . -hat' . -'igm ■ 

 c= b^^ + aT)'^ . . . -2h^7) 



f=-/f . . -av^+h^^ +g^v 



g= . -gr)^ . +hv^-b^^ +fh 



h= . . -h^ +gvUf^^ -c^v- 



And it is to be remarked that we have identically 

 a| + h7?-Hg?=0 

 h^+b7? + f?=0 



Hence of the six equations, a = 0, b = 0, c = 0, f=0, g=0, h=0, 

 any three (except a = 0, h=0, g = 0, or h = 0, b = 0,/=0, or 

 g=0, f=0, c = 0) imply the remaining three. 



* Communicated by the Author. 



