1818.] On Reversion of Series. 109 



y = -== J, (V — 1 a) = x a x 3 + b x b — ex 7 + dx 9 — ex u + &c. 



All these, it is obvious, are merely distinct varieties of the 

 general equation 4- « -1 4 1 a x = x, arising from interpreting a x 

 by or, or — .r, orx V-l. To solve this equation, we first accom- 

 modate it to the circumstances of a particular solution, by making 

 4- x = J- a x, which reduces the equation to 



{fa a ~ x fa a x = )/* a* X = X, 

 and gives the particular values 



x= {^y=)f x y 

 and y = (a~ l $ ax =) a~ l fa-x 



Next, having found an example which satisfies these condi- 

 tions, we proceed to generalize the solution, by assuming 4- x 

 = <p~ l f<p ax, which gives 



(<p~ l f<P a a~ l <p~ x f <p a a x = ) <p~* f°~ f a* x = X 



and S> " * J/**? . 



ly = a <p j ? ** X 



Since « a x = x, whether a.r be 1 or — x, our first two cases 

 both merge into the formula^' 9 x = x, of which a very compre- 

 hensive particular solution is 



/a — b x 

 x = 7 

 b + c x 



[See my former paper]. Hence we immediately deduce for 



Case I. — General Solution. 

 — > /■ _ . — 1 " — 1> * y 



r J r J r j + c ■? 5 

 — i/. _, o — b $ x 



v ■=. <b f a x = <s 7 



Case II. — General Solution. 



\ — b <;>(— .v) 



x = <p '/? -3/ =f 



y =s — « _1 fax = <p 



6 -t- cp (,— .y) 

 _, a — i <? x 



Any exemplification whatever of these very simple cases will 

 be regarded by adepts as superfluous. They will, therefore, per- 

 mit me, as their information is out of the question, to pursue a 

 method of illustration which will have the advantage of render- 

 ing the principles of the functional calculus quite intelligible to 

 those who have not previously made it an object of attention. 

 I start from the most elementary example of Case I. that can 

 possibly be given, viz. 



x ■= \ — y and .'. y = 1 — x 



To generalize this solution : — Select any formula containing y 

 and not r. Assume it to be = z, and thence Jind the value of 



