METHOD OP TRANSFORMING AND RESOLVING EQUATIONS. 347 



A' = 0, B' = 0, C = 0, D' = 0, (330.) 



of the first, second, third, and fourth degrees, by the ratios of 

 in disposable quantities, Oj, a^, . . a^. In like manner it is 

 shown by the result 



m(l, 1, 1, 1, 1) = 47, .... (327.) 



that Mr. Jerrard's general method would not avail to satisfy 

 the five conditions 



A' = 0, B' = 0, C = 0, D' = 0, E' = 0, . . (331.) 



and so to take away^ve terms at once from the equation of the 

 ?nth degree, without any elevation of degree being introduced in 

 the eliminations, unless m be at least = 47, that is, unless the 

 equation to be transformed be at least of the 47th degree ; and 

 the result 



7w(l, 1, 1, 1, 1, 1) = 923, .... (328.) 



shows that the analogous process for taking away six terms at 

 once, or for satisfying the six conditions 



A' = 0, B' = 0, C = 0, D' = 0, E' = 0, F = 0, . (332.) 



is limited to equations of the 923rd and higher degrees. 

 Finally, the result 



m (1,0, 1,1) = 7, (316.) 



and the connected result 



m(l,0, 1,0, 1) = 7, (333.) 



show that it is not in general possible to satisfy, by the same 

 method, a system of three conditions of the first, third, and 

 fourth degrees, respectively, such as the system 



A' = 0, C = 0, D' - a B'2 = 0, (334.) 



nor a system of 3 conditions of the first, third, and fifth degrees, 



A' = 0, C = 0, E' = 0, (335.) 



unless m be at least = 7 j which illustrates and confirms the 

 conclusions before obtained respecting the inadequacy of the 

 method to reduce the general equation of the fifth degree to 

 De Moivre's solvible form, or to reduce the general equation of 

 the sixth to that of the fifth degree. 



[21.] Indeed, if some elevation of degree be admitted in the 

 eliminations between the auxiliary equations, the minor limit 

 of the number m may sometimes be advantageously depressed. 

 Thus, in the process for satisfying the system of equations (330), 

 we first reduce the original difficulty to that of satisfying, by the 

 ratios of ;« — 1 quantities, a system containing three equations 



