Sir William Rowan Hamilton on Equations of the Fifth Degree. 375 



But, for the same equation (147), with the arrangement of the roots (192), we 

 find, by similar calculations, the values : 



H, = - 2-^ 3-' 5^ (10975 - 1472D) ; | 



h,3=-2-^3->5*(10975 + 1472d); J 



and with the arrangement (198), 



h, = -2-^3-'5^(10975 + 3832d); | 



H,3 = — 2-'-3-'5^(10975-3832d). | 



We see, therefore, that in this example, the difference of the two quantities 

 H, and H,3 is neither equal to zero, nor independent of the arrangement of the 

 five roots of the equation of the fifth degree. However, it may be noticed that 

 in the same example, the sum of the same two quantities h, and h,3 has not been 

 altered by altering the arrangement of the roots ; and in fact, by the method of 

 the 43rd article, we find the formula : 



(244) 



"5" V^7 "T H13) = (^2345 X5432) -\- (X2453 X3^.2) -j- (X2534 ^43Si) 



"T (X3254 X4523) + (X4235 X3324^ + (,'^5243 ^3426^ 



I (,X2354 X4532) -J- (X2543 X3452) + (,X2435 X5342) 



r i,X3245 X5423^ "T (.X5234 X4325^ + (X4353 X3J24^ 



of which the second member is in general a symmetric function of the five roots, 

 and gives, in the case of the equation (147), by (221) and (222), the following 

 numerical value, agreeing with recent results, 



H, + H,3 = — 2-' 3-' 5" 439. (245) 



47. It seems useless to add to the length of this communication, by enter- 

 ing into any additional details of calculation : since the foregoing investiga- 

 tions will probably be thought to have sufficiently established the inadequacy of 

 Professor Badano's method* for the general solution of equations of the fifth de- 

 gree, notwithstanding the elegance of those systems of radicals which have been 

 proposed by that author for the expression of the twenty-four values of Lagrange's 



* Professor Badano's rule is, to substitute, in each h, for each power of x', the fifth part of the 

 sum of the corresponding powers of the five roots, x',.,x^ ; and he proposes to extend the same 

 method to equations of all higher degrees. 



