METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 313 



used to determine the 2 m + 3 ratios of the 2 m + 4 quantities 

 9o> '" ?»-2> PPm-i, P'o, - P'm-i^ M', M", M'", M^^, and 

 consequently to give 



^ {x) denoting a known function. The four conditions (113) 

 may next be combined with the 2 m equations (94) and (99), so 

 as to determine the 2 m + 4 ratios of the 2 m + 5 quantities 

 9'o, ... 9'n.-i,p'Pm-i,P"o, ...y'^_,, N', N", N'", N^v^ Nv. 

 and thus we shall have 



N' x"' + N" x'" + N'" x'"' + W x''"' + N^ x''' = Ww (x), (117.) 



the function w (x) also being known; so that, at this stage, 

 the expression (75) for^ will be reduced to the form 



3/ =/(^) = A'" <p (x) + M^v ^ (^) + N^o, (^), . . (118.) 



the three functions <f) (x), \J; (a;), w (x) being known, but the 

 three coefficients A'", M^^, N^, being arbitrary. The condition 

 (114) will next determine the ratio of any one of the quantities 

 9o> '" 9m-2 to anyone of the quantities q'^, ... q'^_2, and 

 therefore also the connected ratio of M^^ to N^, and conse- 

 quently will give 



M^''^{x) + -N^c{a;) = -N^'xi^), ■ ■ • (119.) 

 X (x) being another known function ; and thus we shall have 

 accomplished, in a way apparently but not essentially different 

 from that employed in the foregoing article, the first part of Mr. 

 Jerrard's process, namely, the discovery of an expression for y, 

 of the form 



y=f{x) = A'" <^ {x) + Wx{^), ... (89.) 

 which satisfies the two conditions 



A' = 0, C' = 0, 

 the functions 4) [x) and x (^) being determined and known, but 

 the multipliers A'" and N^ being arbitrary : after which it will 

 only remain to perform the second part of the process, namely, 

 the determination of the ratio of A"' to N"^, so as to satisfy the 

 remaining condition 



D' - a B'2 = 0, 

 by resolving a biquadratic equation. 



[8.] The advantage of this new way of presenting the first part 

 of Mr. Jerrard's process is that it enables us to perceive, that if 

 we would avoid the case of failure above mentioned, we must in 

 general exclude those cases in which the ratios 



