(161.) 



320 SIXTH REPORT— 1836. 



the five ratios of the five first to the last of the six quantities 

 (152), so as to satisfy the five conditions 



•^0,0,1,0 = 0> "] 



B'iAi.o+B'o,i,i.o = 0, j 



B'o,oAo=0^ V. . . . (160.) 



^2,0,1,0 + C 1,1,1,0 + C'o,2,i,o = Oj 



^ 1,0,2,0 + C 0,1,2,0 = ; J 



the seven ratios of the seven first to the last of the eight quan- 

 tities (153), so as to satisfy the seven conditions 



■^0,0,0,1 = 0, 



B 1,0,0,1 + -"0,1,0,1 = 0, 



B'o,0,l,l = <^J B'ooo 2 = 0, 



*^ 2,0,0,1 + ^-^ 1,1,0,1 + C'o,2,0,l = 0, 



C 1,0,1,1 + ^0,1,1,1 = 0, 



^ 1,0,0,2 "^ ^ 0,1,0,2 = ; 

 and the ratio of the last of the quantities (152) to the last of 

 the quantities (153), so as to satisfy the condition 



C'o,o,3.o + C'o,o,2,i + C'o,o,,,2 + C'o,o,o,3 = : • • • (162.) 

 all which determinations could in general be successively ef- 

 fected, without its being necessary to resolve any equation of a 

 higher degree than the fourth. The first part of the process 

 would be now completed ; that is, the assumed expression (149) 

 for y would be reduced to the form 



y=/(a?) = Miv^(^) + SVi";)^(a7), .... (163.) 



the functions <p (a?) and x (^) being determined and known, but 

 the multipliers M'^ and a^'" being arbitrary, and this expression 

 (163) being such as to satisfy the three conditions (8) (10) and 

 (14), 



A' = 0, B' = 0, C = ; 



nineteen out of the twenty ratios of the twenty-one coefficients 

 (150) (151) (152) (153) having been determined so as to satisfy 

 the nineteen equations (157) (158) (159) (160) (161) (162), into 

 which those three conditions had been decomposed. And the 

 second and only remaining part of the process would consist in 

 then determining the remainhig ratio of M'^ to S^'"^, so as to 

 satisfy the remaining condition 



D' = 0, , . . (17.) 



