494 Mr. G. B. Jerrard on a Method of Transforming Eqtiations. 



will be identical. But the question respecting the possibility of 

 these two equations having a common factor of a lower degree 

 than the sixth ought here to be considered. 



We may, it is true, very readily perceive that there cannot 

 generally exist a common factor of either the first or the second 

 degree with rational coefficients, since no cubic radical could 

 under such circumstances be involved in the solution. And 

 when m = 4 (for every other case may be excluded as being 

 already solved by the first method in which X=4)*, it is mani- 

 fest that q and J^, which are each of them expressible as rational 

 functions of the roots of the original equation in x, will neces- 

 sarily lead from the equality of the typical forms to identical 

 equations of the sixth degree ; unless, indeed, there can exist, 

 for equations of the fourth degree as well as for those of higher 

 degrees, a function analogous to D(Vf(.„), Vc^.y), VH(«y))t- 



We see, then, what strong grounds there are for inferring 

 that the equations 



will be identical when 7W = 4. 



But the subject is very instructive, and I hope to return to it 

 at some future time. 



Long Stratton, Norfolk, 

 May 14, 1863. 



Errata in last Number. 

 Page 356, line 4, dele commas in a?^„ . a?^„ . . o^. 

 — 358 — 19,/or P-f Q read P-fQa?. 



* By the mode of solution in which X=4, we can, while m > 4, always 

 effect the proposed transformation by means of an equation of a lower 

 degree in x than the given one ; but when »i=4, the equations 

 a?^+A3j;+A4=0, 

 Ta?^+ Sa^+ Itp»+ Qa?+ P— y=0 

 rise to the same power of a?, and one of them becomes a multiple of the 

 other, as I have shown in my Mathematical Researches. In order, there- 

 fore, to include this case, X was taken equal to 3. 



The method in which X=3 may, howeVer, be made to extend to high 

 values of m. But the equation of the sixth degree in q and that in j^ will 

 no longer be identical, except for particular values of A3, A4, . . Am . This 

 is obvious. 



t See my " Notes on the Resolution of Equations of the Fifth Degree " 

 in the Philosophical Magazine for Februar} .1852: 



