506 Mr. J. Cockle's Analysis of the Theory of Equations, 



in the theory of equations to be induced to follow up the in- 

 vestigation, shall have prosecuted the subject, to such an ex- 

 tent as to be enabled to pronounce a decisive opinion upon 

 the question of the limits of the last-named method. 



26. The subject of equations has, however, lately been re- 

 garded with an apathy which is, to say the least, extremely 

 singular, when we consider its vast range, its importance, and 

 the interest which it undoubtedly possesses in itself. The 

 powers of that great indeterminate process, to the advance of 

 which the labours of Mr. Jerrard have given a fresh impetus, 

 remain as yet untried in practice, and its resources unfathomed 

 even by those who, from its exhaustless springs, might draw 

 the means of irrigating fields of science heretofore deemed 

 barren. The indeterminate methods, when considered in re- 

 ference to the theory of equations, strictly so called, have 

 either completely changed, or presented us with the means of 

 so changing, the whole face of the science. The indetermi- 

 nate theory is, if I may use the expression, one of the Senses 

 of Algebra. 



27. But, even already, the mind has, presented to it, another 

 object than that of merely threading the endless pathways of 

 the indeterminate theory, and of glancing down its illimitable 

 vistas. The indeterminate methods must be compared with 

 one another as wholes, and their mutual relations thoroughly 

 investigated. Of the three general methods, which I have 

 here adverted to, I am disposed to think that the substantial 

 limits are the same. Thus, two simultaneous tertiary quadra- 

 tics, which are immediately solvible by the method of vanishing- 

 groups, may be rendered amenable to the other methods (of 

 vanishing coefficients, and of homogeneous elimination,) by 

 means of transformation or decomposition. Is the method of 

 vanishing groups then the simplest form of indeterminate pro- 

 cess? Or is there an altogether different method, of which 

 the three named above are particular forms, to which they 

 (and all others that may be discovered) can be reduced, and 

 in which all others are included ? To answer this question, 

 definitely and satisfactorily, would be to do, for the indeter- 

 minate theory, that which Lagrange did for the ordinary 

 theory of equations, when he showed the ultimate identity of 

 all the apparently isolated methods, then known, of solving 

 equations of the first four degrees, and the common basis upon 

 which they all rest. 



28. You will forgive me, if I suggest a route to this general 

 indeterminate method, and, to this end, coin a term, adlimi- 

 ?iatio?i, to express an operation by which a quantity is intro- 

 duced into any system of equations. Adlimination and Elimi- 



