330 SIXTH REPORT. 1836. 



m quantities (192.), will in general conduct to null values for all 

 those quantities, that is, to the expressions 



a, = 0, a, = 0, . . . «^ = 0, (216.) 



and therefore to a result which we designed to exclude ; because 

 by the enunciation of the original problem it was by the m — \ 

 ratios of those m quantities that we were to satisfy, if possible, 

 the equations originally proposed. The same excluded case, or 

 case of failure (216.), will in general occur when the solution of 

 the second auxiliary problem gives ratios for the in auxiliary 

 quantities (205.), which coincide with the ratios already found in 

 resolving the first auxiliary problem for the m other auxiliary 

 quantities (204.) ; that is, when the two first problems conduct 

 to expressions of the forms 



a!\ = a a\, a"^ = a a'g, . . . a\ = aa'^, ... (217.) 



a being any common multiplier ; for then these two first pro- 

 blems conduct, in virtue of the definitions (197.), to a determined 

 set of ratios for the m original quantities (192.), namely. 



l,...^2LZLi = !l^3 ..... (218.) 



and unless these ratios happen to satisfy the equation of the 

 third problem (214), which had not been employed in determining 

 them, that last homogeneous equation (214.) will oblige all those 

 m quantities (192.) to vanish, and so will conduct to the case of 

 failure (216). Now although, when the condition (215) is satis- 

 fied, the first auxiliary pi'oblem becomes indeterminate, because 

 m — 1 > ^1 + Aj + A3 + . . . + A< — 1, 



so that the number m — I of the disposable ratios of the m 

 auxiliary quantities (204) is greater than the number of the ho- 

 mogeneous equations which those m quantities are to satisfy, 

 yet whatever system of m — 1 such ratios 



a'l a' 2 "-'m - 1 



(219.) 



we may discover and employ, so as to satisfy the equations of 

 the first auxiliary problem, it will always be possible to satisfy 

 the equations of the second auxiliary problem also, by employ- 

 ing the same system of m — 1 ratios for the m other auxiliary 

 quantities (205), that is, by employing expressions for those 

 quantities of the forms (217); and, reciprocally, it will in ge 

 ncral be impossible to resolve the second auxiliary problem 



