METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 311 



and the excluded case, or case of failure, will now be the case 

 when the sums ^o + P"o^ P'l + ;^"i> — P'm-i + P"m-i are 

 proportional to /?o) Pi> ••• Pm-i^ ^^^^^ ^^> Avhen 



9o + q'o = 0,q,+ q\ = 0, ... q^_^ + j'„,_2 = 0. . . (102.) 



Indeed, it is here tacitly supposed that jo„(_i does not vanish; 

 but Mr. Jerrard's method itself supposes tacitly that at least 

 some one, such as ;?•, of the m quantities p^, ... P„i_ij is dif- 

 ferent from 0, and it is easy, upon occasion, to substitute any 

 such non-evanescent quantity p^ for p^_i, and then to make 

 the few other connected changes which the spirit of this discus - 

 sion requires. 



The expression (101) for f {x), combined with the law (5) of 

 the composition of the coefficients A' and C, gives, for those 

 coefficients, expressions of the forms, 



A' = (1 + jD + ;/) A"i,o,o + A"o,,,o + A'Vn (103.) 



and 



c'=(i +7.+;y)«cVo+(i +;^+yr(c\i,o+c%,o] 



+ (1 + jt> + p') (C"i,,,o + C"i,i,i + C\o,2) X104.) 



4- r," 4. P" 4. P" 4- P" 



' ^ 0,3,0 "^ ^ 0,2,1 ^ ^ 0,1,2 1^ ^ 0,0,3? J 



A"^ ■ ^. and C";^ ^ ^ being rational and integral functions of the 

 3 m - 2 quantities ;jo,^Ji, ...iJ^_i, go, 9i, — 9^-2, g'o' I'u — 

 9'm—2> which functions are independent of ]), p', and L, and are 

 homogeneous of the dimension h with respect to Pq, ... p^_\i 

 of the dimension i with respect to q^, ... q„i_2y ^"'^ ^^ *''^ ^^~ 

 mension k with respect to q'^, ... q'm-2'> ^^^V '"^^^ '^^^^ such 

 that the sums 



A"o,,,o + AV,i (105.) 



and 



C\i,o + C%,, ...... (106.) 



are homogeneous functions, of the 1st dimension, of the m — 1 

 sums q^ + q'o, ... q„^_^ + q'm-2> ^^hile the sum 



C%,o + C"i,i,i + C"i,o,2 (107.) 



is a homogeneous function, of the 2nd dimension, and the sum 



C'o,3,0 + ^"o,2,l + ^"o,l,2 + C 0,0,3 * ' • (108.) 



is a homogeneous function, of the 3rd dimension, of the same 

 m — 1 quantities. These new expressions, (103) and (104), for 



