348 SIXTH REl'ORT — 1836. 



of the first degree, two of the second, and one of the third ; and 

 we next reduce this difficulty to that of satisfying, by the ratios 

 of m — 2 quantities, a system containing five equations of the 

 first, and two of the second degree. Now, at this stage, it is 

 advantageous to depart from the general method, and to have 

 recourse to ordinary elimination ; because we can thus resolve 

 the last-mentioned auxiliary system, not indeed without some 

 elevation of degree, but with an elevation which conducts no 

 higher than a biquadratic equation j and by avoiding the addi- 

 tional decomposition which the unmodified method requires, we 

 are able to employ a lower limit for m. In fact, the general 

 method would have led us to a new transformation of the ques- 

 tion, by which it would have been required to satisfy, by the 

 ratios of m — 3 new quantities, a system containing six new 

 equations of the first, and one of the second degree ; it would 

 therefore have been necessary, in general, in employing that 

 method, that ?n — 3 should be greater than 6 + 1, or in other 

 words that m should be at least equal to the minor limit eleven ; 

 and accordingly we found 



m (1,1, 1,1) = 11. . . . (317.) 



But when we dispense with this last decomposition, we need 

 only have m — 2 > 5 + 2, and the process, by this modifica- 

 tion, succeeds even for m = ten. It was thus that Mr. Jerrard's 

 principles were shown, in the tenth article of this paper, to 

 furnish a process for taking away four terms at once from equa- 

 tions as low as the tenth degree, provided that we employ (as 

 we may) certain auxiliary systems of conditions, (160) and (161), 

 of which each contains two equations of the second degree, but 

 none of a degree more elevated. But it appears to be impos- 

 sible, by any such mixture of ordinary elimination with the ge- 

 neral method explained above, to depress so far that lower limit 

 of m which has been assigned by the foregoing discussion, as to 

 render the method available for resolving any general equation, 

 by reducing it to any known solvible form. This Method of 

 Decomposition has, however, conducted, in the hands of its in- 

 ventor Mr. Jerrard, to several general transfurmatiojis of equa- 

 tions, which must be considered as discoveries in algebra j and 

 to the solution of an extensive class of problems in the analysis 

 of indeterminates, which had not before been resolved : the 

 notation, also, of symmetric functions, which has been employed 

 by that mathematician, in his published researches* on these 

 subjects, is one of great beauty and power. 



• Mathematical Researches, by George B. Jerrard, A.B., Bristol; printed 

 by William Strong, Clare Street ; to be had of Longman and Co., London. 



