346 SIXTH REPORT — 1836. 



= 1 + 7«(A, + 7^2 + . . + ^^ _ j,^2 + ..+^t-V"^'t-i)>l 



so that, for example, 



7n(l,l,l,l,l) = 1 +m(4,3,2,l); (323.) 



m (4, 3, 2, 1) = 1 + m (9, 5, 2) = 46 ; (324.) 



m(l,l,l,l,l,l) = l +m(5, 4,3,2,1); . . . (325.) 

 ?n(5,4,3,2, 1) = 1 + m(14,9,5,2) = 922; . . (326.) 



and therefore 



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



and 



m (1,1, 1,1, 1,1) = 923 (328.) 



[20.] The formula 



w(l,l,l) = 5, (307.) 



may be considered as expressing, generally, that in order to 

 satisfy a system of three homogeneous equations, rational and 

 integral, and of the forms 



A' = 0, B' = 0, C = 0, (329.) 



that is, of the first, second, and third degrees, by a system of 

 ratios of m disposable quantities 



a„...a^, (192.) 



which ratios are to be discovered by Mr. Jerrard's method of 

 decomposition, without any elevation of degree by elimination, 

 the number m ought to be at least equal to the minor limit ^i;e; 

 a result which includes and illustrates that obtained in the 4th 

 article of the present communication, respecting Mr. Jerrard's 

 process for taking away three terms at once from the general 

 equation of the mth degree : namely that this process is not gene- 

 rally applicable when m is less than /^we. Again, the process de- 

 scribed in the eleventh article, for taking away, on Mr. Jerrard's 

 principles, four terms at once from the general equation of the 

 Tilth degree, without being obliged to eliminate between any two 

 equations of condition of higher degrees than the first, was 

 shown to require, for its success, in general, that ?n should be 

 at least equal to the minor limit eleven ; and this limitation is 

 included in, and illustrated by, the result 



»i(l, 1, 1, 1)=11, .... (317.) 

 which expresses generally a similar limitation to the analogous 

 process for satisfying any four homogeneous equations of con- 

 dition, 



