312 



SIXTH REPORT — 1836. 



the coefficients A' and C, must identically coincide with the 

 former expressions (83) and (84), when we return from the pre- 

 sent to the former notation, by changing p, p\ p^, p^, ... 

 Pm-v 5'05 9i, — ^m-Qy ?'o> ^'iJ •- 9'm-2' ^o their values as 

 functions of A', A", A'", M', M", M'", M'^, N', N", N"', N^^ N^ ; 

 and hence it is easy to deduce the following identical equations : 



-^ 1,0,0 = -^ 1,0,0 5 I 



A'o,i,o=i?A%^o + A"o,,,o; V (109.) 



-A 0,0,1 = /> A j^o^o 4- A o^o^i ; J 



and 



P' — cii • ^ 



^ 3,0,0 — ^ 3,0,0 > 



*^ 2,1,0 = op L> 30Q 4- C 2j^o5 



*^ 2,0,1 =op\^ 300 + C 2 0,1 ; 



C',,.,0 = 3p^ C%,o + 2p c\i,o + C",,2,o ; 



C'm,, = 6;,^'C"3,o,o + 2y CV,o + 2^C%,i + C\,,,; j^ ^^^^^^ 



^ 1,0,2 = ^i*^ C'3,0,0 + 2/)' C"2,o,i + C"i^o,2 J 



^0,3,0 + ^0,2,1 + ^'0,1,2 + ^0,0,3 — iP + P^^ 3,0,0 



+ {p+pr{c\uo + c\o,^) 



+ (2>+y)(C\2,o+C%,, + C%,2) 



+ C 0,3,0 + ^ 0,2,1 + ^ 0,1,2 + ^ 0,0,3- 



The system of the ten conditions (85), (86), (87), (88), may 

 therefore be transformed to the following : 



. (111.) 



. (112.) 



. (113.) 



• (114.) 



■^ 1,0,0 — ^} ^ 3,0,0 — ^ } 



A 0,1,0 ^ 0, L- 2,1,0 = 0, iu 1^2,0 ^^ 5 . • . • 



\ii — o c," — C." — n C" — O • 



^ 0,0,1 — "Jj ^ 2,0,1 — ^•'j ^ 1,1,1 — 'Jj ^ 1,0,2 — " J 



C 0,3,0 + C 0,2,1 + ^ 0,1,2 + ^ 0,0,3 = 5 • • 



and may in general be treated as follows. The two conditions 

 (111) may first be combined with the m equations of definition 

 (46), and employed to determine the m + 2 ratios of the m + 3 

 quantities /)o, -"Pm-ii ^'> ^"y ^"' i ^^^^ therefore to give a re- 

 sult of the form 



A' x"-' + A" x"-" + A'" x"-'" = A"' <p (x), . 



(115.) 



the function (p {x) being known. The three conditions (112), 

 combined with the 2 m equations (47) and (56), may then be 



