239 

 for the quartic, it is 



(^-i)(p-^)(p-'!) ,, 



^ D(D — 1)(D— 2) ^ ^~-^' 



and for the quintic, it is 



^ D(D — 1)(D— 2)(D— 3) ^ ^-^' 



which are all comprehended under the general form given 

 on page 199. The induction, though incomplete, is yet 

 sufficiently wide for the purposes of the present theory, inas- 

 much as, when 7i is greater than 5, the given n-ic equation 

 cannot in general be reduced to the trinomial form with which 

 we are now working. But Mr. Cayley's brilliant piece of 

 analysis, of which an abstract is given on page 193, enables 

 us, as we have seen, to establish the theorem in all its 

 generality. 



I noticed in my last commuhication the exceptional case 

 n=z2. The following remarks on the same subject by Mr. 

 Cayley will be read with interest. I had taken the liberty 

 of calling his attention to the form of the resolvent for 

 the quadratic, and replying in a letter to me, under date 

 25th February, 1862, he says, "I ought to have seen, 

 and after writing the note on the Differential Equation did, 

 in fact, see from your paper that n=2 is an exceptional 

 case : the d priori reason is, that for the equation 



we have (not as in the general case yi -}- t/z 'h <&c. = 0) but 

 ^1+^2=0. Hence for n=:2, the differential equation cannot 

 be of the form 



for if so, its solution, which would be of the form y=:CY, 



