344 SIXTH RBPORT — 1836. 



in which, 



*li = ^1 + 1, >)j = A^ + 1, »,3 = //g -f 1 (306.) 



For exumplc, 



m{l,l,l) = 5 (307.) 



[18.] When 



t = 4, (308.) 



that is, when some of the original equations are as high as the 

 fourth degree, (but none more elevated,) then 



'^3 = ^3 + 1^4(^4 -I), 



\ = ^^ + ^4^ + ^ [K + 1) K (^4 - 1), 



+ i(^4 + 2)(7/4+l)//,(>^,_l)j J 

 ''K = \ + yh{h,-\), -I 



'\=^'h, + 'h^'h, + ^i:h., + \Yh,(:h,-i)',] • • ^ "-J 



''\ = ''fh^y\('K-^); (311.) 



and the minor limit of m, denoted by the symbol m (Aj, Ag, ^3, h^, 

 is given by the equation 



m {h„ h^, 7/3, 7/4) = ^,j +^^3 + "7/2 + ''\ + 1 ; . . (312.) 



which may be thus developed, 



m (Ai, Ao, A3, A4) = ,j + -L [,J^ + ^% b-sT ^ 



(309.) 



2 



+4-w+-^M^ 



+ 



H313.) 



+|m-'' b,V + W^{4 W^ +1 W^ 4-^ [,j4| 



+ I [14]* + I [.j** + p [»4]'^ + -^ W V -j^ [>,4]S 



if we employ the notation of factorials, and put for abridgement, 



>), =Ai + 1,....J4='^4+ 1 (314.) 



In the notation of powers, M'e have 



