214 UflLITY OF A SERIES IN FINDING A FLUENT. 



this, I beg to offer you a detenu i nation of a fluent, which 

 seems to have been considered as presenting some difficul- 

 ties scarcely to be overcome. 



The equation to be resolved is this ; fx y x */ [x" 3, -\- y % ) 

 zzz mxj: hence, by squaring both sides, we have [{fxyx)* 



-m*x*)j*+(fxyx)*x*=0, and [(fxy *)* — m % **] |s 

 -f ( fi y *)* =z0. Put y - a ± b .r 2 + c a 4 + d J + . . . , 

 then /a- tf f t TZj ax % + $ b x 4 + ^ c x 6 \- . . .; of which the 

 square may be called Aa ,4 + Ba 6 + C .i 8 -f . . . : again, 



~ -3 2 6 .r 4* 4 c x 2 + 6 r/ x s -\- . . . , the square of which may 



be called a, x* -f # a- 4 " -J- y .» 6 -f- . . . ; then, by multiplication 

 and substitution, our equation incomes (A — * m*j * 4 + 

 (A * 4- 1) — I-: m x j .r 6 + (A /3 -f- I> * + C — y »*j x* + . . . 

 rrO; whence, by the general taw of in;iniie series, An: 

 «w a , Aa-r- B = £>»*, A/3 f li* + C = ym% ..., and 



ei zz—z, j3z r •, and so forth; but, by actual invo- 



m nr 



lution, we find A — I a*, B — £ a b, C — £ « c 4- T V b % , . . . , 



and again a = 4 b z , (3 — 16 b c, ? — 24 6 J + 16c 1 ,..., 



, , . * /3 y — l6cc £ — 48cd 



whence 6 = \/ -— , c — -— , a zz — , ezz ., 



4 166 24 32 



and so forth; and finally, by reduction, we have b zz 



4 w 



C = 6 8 +~-, d-2b* + ~+ 1 ^r-^ and ezz5b^ 

 10W 18/H 5?0;w 



35& 5 f 41 6 3 ■ i , . .. . !* 



1- — r 4 — -—rr — = ; and in the case where a be- 



v 32 m J 152 m* 30864 m 3 



comes extremely small, so that its higher powers may be 

 Mglccted. c a jjj, d = g^, « = £-, and/=^. If, 



in particular cases, beside the general equation, we have a 

 determination of the value offxyx, for instance fxy x zz 

 nx, we may obtain an equation between x and a, whence 

 either of ihem may be found from * he other; that is, \ax 

 -f- \ bx* + i c * 5 + • • • — n: hence, if x becomes very 



6mall, a zz.^—\ and if a is evanescent, -pli-f j~t~ 



X 5 x 7 \ 



4 5V. 4* m* + 4. 2\ 4*. <>> m 3 + •'•;=»• - 



For 



HI 



