110 On Reversion of Series. [Feb. 



y in terms pf z. [In functional symbols, make <p y = z ; : y 

 — < r>~ > z -l F° r z > ? " * ne expression of this value, substitute the 

 whole expression which arises from substituting the selected func- 

 tion of y, for y in the right hand scale of the Jirst given equation. 

 The result is a new value of x. For y, in this value, substitute 

 x, and it will give the simultaneous new value of y. 



Ex. 1. — A simple multiple, (<p y =) a y = z .-.?/(= p -1 ) = - 



,,., 1 — a u t I — « x 



Whence x = = and y = 



Ex. 2. — The multiple of a power, a y 3 = z ;. y = \/ ~ a 



3 / 1 — ay* 3 / 1 



Whence x = </ — 7— and y = * / 



1 - a i* 



JEx.3. — A compound quantity, y* + 2y = Z.'.y = */z+l — 1 



Whence x = \(2 — 'ly — y- — 1 and y = V 2 — 2 x — x* — 1 



Ex. 4. — A circular transcendental, sm.y = z .■ . y = arc to sin. z 



Whence x = arc to sin. (\—sin.y), and y = arc to sin. (1 —sin.x) 



Ex. 5. — An exponential, a y — z :.y =■ * ■ 

 Whence x = ~ and y — 



log. o ' Jog. a 



Were these results expanded into infinite series, each pair of 

 series would manifestly exhibit the relation which characterizes 

 Case I. To adapt them to Case II., it is only requisite to sub- 

 stitute — y for y in the values of x, and to prefix the negative 

 sign to the values of y. 



Ex. 3, for instance, would be transformed to 



x = V 2 + 2j/ — y* — 1 and y = — */ 2 — 2 x — x* + 1 



Case III. — The particular equation to be resolved in this 

 case isf* — x = x, or rather, exchanging x for — x, 



f* x = — x, 

 of which the only solution immediately obvious is 



fx = x 4/ — : 1, 

 which is not capable of further extension. 



Recourse must be taken to another artifice. Among the 

 functions of circular arcs which furnish pairs of convertible 

 equations, such as 



sin. y = cos. x .'. sin. x — cos. y, 

 tan. y — cos. x : . tan. x =■ cos. y, 

 there occurs one, which involves an impossibility ; and on endea- 

 vouring to neutralize this by introducing the imaginary coefficient 

 is/ — 1, we pass from the conditions of the first, to those of the 

 third case. The formula f- x = — x seems indeed to indicate 

 such a process, and renders it probable that other solutions may 

 be found on the same principle. 



