326 SIXTH REPORT. — 1836. 



^_^_i = p'l, (184.) 



Pm-\ 

 Po'" - P>0 = W. • • • P"'m - 2 - fPm 2 = i'\n - 2. (185.) 



^'"-^ = y', (186.) 



Vm-2 



"""'-^ = r, (188.) 



^w-3 



V - *• ^^O = 'o» • • • ^»» 4-^^'m 4 = '», 4. • • • (189.) 



and retaining the analogous expresssions (164.) (54.) (55.) (96.) 

 (97.) (124.) (125.), we find, by a reasoning exactly analogous to 

 that employed in the former discussions, that the final expres- 

 sion for y will in general be of the useless form 



y = L X, (190.) 



ill the following case of failure, 



^o = 0,^, = 0,...^^_4 = 0; (191.) 



and on the other hand that the seven conditions (1770 '^^y ^^ 

 reduced to the form of seven rational and integral and homoge- 

 neous equations between these m — 3 quantities t^, t, ... tm — Ai 

 so that the case of failure will in general occur in the employ- 

 ment of the present process, unless the monher m — 3 be greater 

 than seven, that is, unless the degree m of the ])roposed equation 

 in X be at least equal to the minor limit eleven. 



It must, however, be remembered that the less complex pro- 

 cess described in the foregoing article, (since it contained no 

 condition, nor group of conditions, in which the dimension, or 

 the product of the dimensions, exceeded the number four,) agreed 

 sufficiently with the spirit of Mr. Jerrard's general method; 

 and was adequate to take away four terms at once from the ge- 

 neral equation of the tenth, or of any higher degree. 



[12.] The various processes described in the 2nd, 5th, 6th and 

 11th articles of this communication, for transforming the ge- 

 neral equation of the mth degree, by satisfying certain systems 

 of equations of condition, are connected with the solution of this 

 far more general problem proposed by Mr. Jerrard, " to dis- 

 cover m — 1 ratios of m disposable quantities, 



«,, ^2, ...«„„ (192.) 



