342 SIXTH REPORT — 1836. 



'A' = 0, ^A" = 0, . . . VV^") = 0, (288.) 



and one of the second degree, 



'B' = 0, . . . . (289.) 



by the n + 1 ratios of w + 2 disposable quantities, 



«1J «2J •••«« + 2» • • • (290,) 



it is permitted to proceed as follows. Decomposing each of 

 the first n + 1 quantities into two parts, so as to put 



a, = a', + a"„ a^ = a\ + a\, ...«„ + 1 = <+ i + a"„ + i, (291 .) 



we may decompose each of the given functions of the first de- 

 gree, such as 'A^*^, into two corresponding parts, 'A^*^ and'A^*^, 



of which the former, 'A^"). is a function of the first decree of the 

 ' 1,0 ° 



n -\- 2 quantities, 



«'„a'2, ..a'„^l, a'„^2' (292.) 



while the latter, 'A^*), is a function of the first degree of the 

 n -\- \ other quantities 



a!\,a\,..a\^^', ....... (293.) 



and then, after resolving in any manner the indeterminate pro- 

 blem, to satisfy the n equations of the first degree, 



^A'j,0=0,WYo = 0, ...^AW = 0, .... (294.) 



by a suitable selection of the n -\- \ ratios of the n -{■ 2 quan- 

 tities (292), (excluding only the assumption a^^ _|^ g = 0^) we may 

 determine the n ratios of the n + \ quantities (293), so as to 

 satisfy these n other equations of the first degree, 



Wo,l = 0, Wo,i = 0, . . . ^AW = ; . . . . (295.) 



after which it will only remain to determine the ratio of any one 

 of these latter quantities (293) to any one of the former quan- 

 tities (292), so as to satisfy the equation of the second degree (289), 

 and the original problem will be resolved. 

 [17.] Again, let 



< = 3j (296.) 



that is, let us consider a system containing A, equations of the 

 first degree, such as those maiked (193), along with h^ equations 



