Mr. A. Cayley on the Theory of Matrices. 223 



An alio)' of 12 parts of bismuth and 1 part of tin — 13-670 



An alloy of 2 parts of antimony and I part of zinc .... —22-700 

 Tellurium (from M. Alexander Loewe, purified by 



M. Holtzmann) — 1 79-80 



Selenium (from the Collection of the Heidelberg Chemical 



Laboratory) —290-00 



The method by which these determinations were made is the 

 following : — Two thermo-elements, whose warm and cold soldering 

 points had the same temperatures, were compared with each other ; 

 these formed a circuit with the coil of a multiplicator, which sur- 

 rounded a magnet rod (of about a pound weight) to which was fast- 

 ened a piece of looking-glass, thereby allowing the deflections of the 

 magnet to be observed at a distance by means of a telescope and 

 scale, in the same manner as observations are made with the mag- 

 netometer. Two commutators were also brought into the circuit ; the 

 one changed the direction of the current in the wire of the multipli- 

 cator, the other allowed the currents of the thermo-elements to pass 

 either so as to strengthen, or so as to oppose each other. 



The foregoing experiments were carried out in the Physical Ca- 

 binet at Heidelberg, under the direction of Professor Kirchhoff, to 

 whose advice and assistance I am much indebted. 



"A Memoir on the Theory of Matrices." By Arthur Cayley, 

 Esq., F.R.S. 



The term matrix might be used in a more general sense, but in 

 the present memoir I consider only square and rectangular matrices, 

 and the term matrix used without qualification is to be understood 

 as meaning a square matrix ; in this restricted sense, a set of quan- 

 tities arranged in the form of a square, e. g. 

 { a, b, c ) 

 I a , b) c' I 

 I a", b", c" I 



is said to be a matrix. The notation of such a matrix arises naturally 

 from an abbreviated notation for a set of linear equations, viz. the 

 equations 



'^=ax-\-by-\-cz 

 Y = a'x + b'y + c'g 

 Z=a"x+b"y + c"z 



may be more simply represented by 



(X, Y, Z)=( a. 6, c Jx,t/,z) 

 a, b' , c' 

 a", b", c" 



and the consideration of such a system of equations leads to most 

 of the fundamental notions in the theory of matrices. It will be 

 seen that matrices (attending only to those of the same degree) com- 

 port themselves as single quantities ; they may be added, multiplied, 

 or compounded together, &c. : the law of the addition of matrices is 



