314 SIXTH REPORT — 1836. 



y'o g'' ... g'"'-3 /,<,^x 



determined, as above explained, through the medium of the 

 conditions (113), coincide with the ratios 



-i^, -J^,,..l^, (i.l.) 



determined, at an earlier stage, through the medium of the con- 

 ditions (112). In fact, when the ratios (120) coincide with the 

 ratios (121), they necessarily coincide with the following ratios 

 also, 



go + 9'o 9l + g'l 9m-3 + 9'm-3 - 



9m-r'2 + 9'm-2 9m-2 + 9'm-2 '" y,„_2 + 9' m-'l' ' ' ^"'^ 



and unless the ratios, thus determined, of the m — 1 sums 

 go + g'oj ••• 7w— 2 + 'im—ii ^^^ accidentally such as to satisfy 

 the condition (114), which had not been employed in determi- 

 ning them, then that condition, which is a rational and integral 

 and homogeneous equation of the third degree between those 

 quantities, will oblige them all to vanish, and therefore will con- 

 duct to the case of failure (102). Reciprocally, in that case of 

 failure, the ratios (120) coincide with the ratios (121), because 

 we have then 



g'o = - go> g'l = - ?n — g',«-2 = - 9m-i- • • • (123.) 



The case to be excluded, in general, is therefore that in which 

 the in — 1 quantities g'o? '•• ?'m-2 ^^"^ proportional to the m —\ 

 quantities q^, ... 9^_Q.i ^'^^ ^^^^ consideration suggests the in- 

 troduction of the following new symbols or definitions, 



9 m — 2 /t rtj \ 



=<?> (124.) 



9m-1 

 g'o-ggo=^o>g'i-?gi = '-u — ?'m-3-g?m-3=^»^-35 (125.) 



because, by introducing these, we shall only be obliged to guard 

 against the simultaneous vanishing of the m — 2 quantities 

 *'o5 *'i5 ••• ^'tw-S' *^^^* ^^' ^^ shall have the following simplified 

 statement of the general case of failure, 



ro=0, r, = 0, ... r,„_3=0 (126.) 



Adopting, therefore, the definitions (124) and (125), and con- 

 sequently the expressions 



