316 SIXTH REPORT — 1836. 



The three conditions of the group (135) differ only in their no- 

 tation from the three conditions (112), and are to be used ex- 

 actly like those former conditions, in order to determine the 

 ratios of 90, ... 9^_2) after the ratios oi j)o> ••• Pm—\ have 

 been determined, through the help of the conditions (111) ; but, 

 in deducing the conditions (136) from the conditions (113), a 

 real simplification has been effected (and not merely a change of 

 notation) by suppressing several terms, such as q A."'q j q, which 

 vanish in consequence of the conditions (112) or (135). And 

 since we have thus been led to perceive the existence of a group, 

 (136), of four homogeneous equations (rational and integral) be- 

 tween the m — 2 quantities r^, r^, ... rjjj_3,we see, at last, that 

 we shall be conducted, in general, to the case of failure (126), 

 in which all those quantities vanish, unless their number m — 2 

 be greater than four ; that is, unless the degree of the proposed 

 equation in x he at least equal to the minor limit seven. It 

 results, then, from this analysis, that for equations of the sixth 

 and loiver degrees, Mr. Jerrard's process for effecting the trans- 

 formation (20), or for satisfying the three conditions (8) (14) 

 and (19), 



A' = 0, C = 0, D' - « B'2 = 0, 



will, in general, become illusory, by conducting to an useless 

 expression, of the form (93), for the new variable y ; so that it 

 fails, for example, to reduce the general equation of the Jifth 

 degree, 



a;5+A^4^B^+Ca,2 + D^ + E=0, . . (69.) 



to De 3Ioivre's solvible form, 



3,5 + B' ^ + 3- B'^y -f E' = 0, . . . . (137.) 



except, by an useless assumption, of the form 



3/ = L(a?5 + Aa?'»+ Ba^+ Ca?^ + D<r + E), . . (138.) 



which gives, indeed, a very simple transformed equation, namely, 



y' = 0, (139.) 



but affords no assistance whatever towards resolving the pro- 

 posed equation in x. Indeed, for equations of ihe fifth degree, 

 the foregoing discussion may be considerably simplified, by ob- 

 serving, that, in virtue of the eight conditions (112) (113) (114), 

 the four homogeneous functions (105) (106) (107) (108), of the 

 m — 1 sums Jo + ?'o> ••• S'm-i +9'»n-2 5 are all = 0, and there- 

 fore also (in general) those sums themselves must vanish (which 

 is the case of failure (102),) vvhen their number w — 1 is not 



