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XLIII. On the Solution of certain Systems of Equations. By 

 James Cockle, M,A., Barrister 'at-Law, of the Middle 

 Temple^, l 



(1.) npHE following investigations were suggested to me by 

 A the perusal of Mr. Sylvester's paper at pp. 370-373 

 of the last November Number of this Journal ; but they do not 

 involve the theory of determinants. In no spirit of disparage- 

 ment of that theory^ nor of the splendid scientific achievements 

 of Mr. Cayley with reference to it^ I venture to intimate an 

 opinion that_, as the theory of determinants (in its explicit form 

 at least) is in no degree indispensable to the progress of the 

 theory of algebraic equations,, so also that its processes have no 

 decided superiority over others that enter into the algebraic 

 theory. This introductory remark must be considered in exclu- 

 sive reference to the theory of equations, otherwise it would be 

 indicative of impertinence on my part, as well as of inaccuracy. 

 (2.) Let there be given for solution m simultaneous equations. 

 And, further, suppose that, by some of the known artifices of 

 algebra, those m equations can be put under the respective forms 



Mj -f = 0, ^2 + 11 = 0, ..., 1^^+11=0, . . {a.) 



then the solution of the system («.) involves that of the given 

 equations. 



(3.) Let u^—Ur+i = v^, 



then, if we can satisfy the m relations 



we can satisfy the system («.). We might give various forms to 

 Vr, and consequently to (b.), but I have selected that which ap- 

 pears to be the most convenient. These forms may however be 

 departed from as individual examples may render it desirable. 



(4.) Let a: be one of the unknowns involved in the given 

 equations. Then, if we assume that 



where A^ either is free from x or capable of being made to vanish 

 without determining iCj we obtain a very remarkable form of Vr ; 

 for, in this case, the solution of (b.) reduces itself to that of the 

 system 



Ai=0, A2=0, . . . , A^.,=0, u^ +n=0. . (c.) 



* Communicated by the Author, who adds the following note : — 

 1 [" There are one or two observations which I should have been glad to 

 have included in the above paper. But, as it has already extended to the 

 limits within which it is perhaps desirable that I should confine myself, I 

 shall seek another opportunity of laying them before the readers of this 

 Journal.— James Cockle."] 



