Mr. J. Cockle on Symbolical Decomposition, 289 



1 d^a 

 and that on eliminating - • j-% from (4) by means of the relation 



1 cPu 2q du 



u dx* a. dx ' 



which; since « is supposed to be a particular integral of (2), is 

 seen to be satisfied, we are led to a result substantially identical 

 with (3). I may add that, if (2) and 



(£-+■*= +'>-.° (5) 



have a common solution, a, then, a not vanishing, 



s=r 

 necessarily. 



I have applied the same process to a linear differential equa- 

 tion of the third order. Its application, however, seems to 

 require an artifice needless in the decomposition of equations 

 of the second order. 



2. If it be thought needful to put M. Hermite's final result 

 under the form indicated in art. 10 of Mr. Jerrard' s " Supple- 

 mentary Remarks," &c, it must also be borne in mind that N 

 is a function of roots of the reduite, and that the case put by 

 Mr. Jerrard in art. 11 is (in Mr. Jerrard' s point of view) ex- 

 plained by the fact that in such case the irreducible quintic 

 radicals are contained in the roots of the reduite. However, 

 although Mr. Jerrard has conceded, I think more than once, 

 that the roots of a general quintic cannot be expressed without 

 quintic surds, yet he is no doubt entitled to adopt, for the pur- 

 poses of arts. 12 and 13, the hypothesis that the roots can be 

 expressed in terms of cubic and quadratic radicals only. But 

 why is this hypothesis shifted in art. 14 ? The hypothesis being 

 supposed true, Mr. Jerrard, in art. 13, concludes that 2) ought 

 to be different from zero. The hypothesis being supposed false, 

 Mr. Jerrard, in art. 14, is forced to come to the conclusion that 

 2) ought to be equal to zero. The antecedents of these hypo- 

 thetical propositions are contradictory ; and though we may, by 

 combining them, arrive at fche result that, if the hypothesis be 

 supposed to be simultaneously true and false, X) ought to be 

 simultaneously different from and equal to zero, yet such a 

 result is nugatory, and doe3 not establish any incongruity or 

 error whatever in M. Hermite's argument. Mr. Jerrard' s argu- 

 ment being invalid, it is unnecessary to inquire whether he is 

 justified in assuming that the denominator of an expression in- 

 dicating an impossibility is zero. 



3. I would rather be regarded as a student than a critic of 



