322 SIXTH REPORT 1836. 



connected by a relation of the form (36), we must avoid, as 

 in the discussion given in the seventh article, the case where the 

 m svimsp'Q+p"Q,...p'^_x + p"m-\ are proportional to the m 

 quantities /?<,, ... Pm-u that is, the case 



?o + Q'o = 0, ... ?^_2 + q'm-a =0, ... (102.) 



if we adopt the definitions (54) (55) and (96) (97), so as to in- 

 troduce the symbols p, q^, q^, ... q^^^^ and^, j'q, q\,"'q'm-2 • 

 With these additional symbols it is easy to transform the condi- 

 tions (160) into others, which (when suitably combined with the 

 equations of definition, and with the ratios of p^, ... /'^_i al- 

 ready previously determined through the help of the conditions 

 (157) (158) (159),) shall serve to determine the ratios (121) of 

 S'oj ••• ?7n-2 5 ^"^ *hen to determine, in like manner, with the 

 help of the conditions (161), the ratios (120) of j'q, ..'q'^^^'i 

 after which, the condition (162) may be transformed into a ra- 

 tional and integral and homogeneous equation of the third degree 

 between the sums q^ -f- q'^, ... 9^_2 + 9'm-'i> and will in gene- 

 ral oblige those sums to vanish, if their ratios (122) have been 

 already determined independently of this condition (162), which 

 will happen when the ratios (120) coincide with the ratios (121), 

 that is, when the quantities q^, ... q'^_\ are proportional to 

 the quantities go, ... 5',„_i • We must, therefore, in general 

 avoid this last proportionality, in order to avoid the case of 

 failure (102) ; and thus we are led to introduce the symbols 

 ffj **o> *'iJ ••• ^M-S'*^^^"*^^ hy the equations (124) (125), and to 

 express the case of failure by the equations 



,-0 = 0, r, = 0, ... 7V«_3=0. . . . (126.) 



With these new symbols we easily discover that the seven con- 

 ditions (161) may be reduced to seven rational and integral and 

 homogeneous equations between the quantities »•(,, rj, ... »*„j_3, 

 which will in general oblige them all to vanish, and therefore 

 will produce the case of failure (126), unless the number m — 2 

 of these quantities be greater than the number seven, that is, 

 unless the exponent m of the degree of the proposed equatioji be 

 at least equal to the minor limit ten. It results, then, from 

 this discussion, that the process described in the present article 

 will not in general avail to take away four terms at once, from 

 equations lower than the, tenth degree, and, of course, that it 

 will not reduce the general equation of the fifth degree^ 



a?^ + A a?" + B a^ + C .z"2 + D a- + E = 0, . . (69.) 



