52 Mr. Cockle on Equations of the Fifth Degree. 



the quantities /' and I", being perfectly arbitrary, may be sup- 

 posed to satisfy not only the equation (c.) but also the fol- 

 lowing, 



A"'^f' + Aiv;:'^=/'.t'f + /".rr, ... (c'.) 



where .r„ is a root other than .r„. Let Xa be .«•„_„ then 



l«-i=:0; . (e'.) 



and we must write [l,...l„_2]^ instead of [],...l„_,]^ in 

 equation (f.). But in this case, in order that we may satisfy 



[i,...U.?=o (g'.) 



without making the I's vanish, we have the condition 



n-2> 1, 



or ?z = 4- at least; and ^Y may be of such a form as to render 

 it necessary that 7i be not less than 5, which is the case in the 

 transformation above considered. As to this, see page 395 of 

 the last-mentioned volume, where I have pointed out the error 

 here discussed, but not in sucli a manner as to make the fore- 

 going explanation superfluous. 



III. As connected with, and by way of concluding, this 

 part of the subject, let 



then (1.) gives 



which will be of the form 



M'HDtt-' + D^ = 0, (4.) 



provided that 



Now let 'ay'=;^(D) 



be a solution of (4.), then 



will also be a solution: by combining these solutions we ob- 

 tain the functional equation 



X^(«;') = ^^'', orx'(D) = D; 



atid the same result might be obtained from any trinomial 

 equation of the fifth degree — as I have already noticed (vol. 

 xxviii. p. IS."], note). 



IV. At pages 190, 191 of the same (28th) volume of the 

 present work, there occur errors resembling those at pages 



