420 Mr. Herapath on the Solution of ^x = x. [Dec. 



Article III. 

 On the Solution of -\> n x = x. By John Herapath, Esq. 



(Concluded from p. 329.) 



§ 4. If in our primitive equation $," x = x we assume x = u 2 

 and ^' x = ?/. + ,, q being any quantity, and u a function to be 

 determined, there results 



«x + n = V x = a? = ?<, 



z 



and by integration ?/. = 1" x A, 

 A being the arbitrary constant. But because A is any quantity, 



and 1" is a well known function of sin and cos it is 



« n 



2 k X z 



also a function of sin or cos ; and therefore 



U, = 1" x A =/ (l' 7 ) = psin 1^, 



where f is an arbitrary function, and consequently <p. Again 



since 



. 2kXz , , . . 2 ft x (« + v\ 



x = u, = <p sin , and ^ x = u 2 + v = <p sin ' 



we have 



81 + v — 1st-. — sin -1 <s~ l x + v 



2 k X z ~ 



and 4/"* = <p sin 5 — h sin - ' <p -1 x£ (25) 



a solution, with the advantage of being direct, and considerably 

 more simple, is equally as general as either of the former. For 

 instance, in the case of n = 2, if v = 1 and k any odd number, 



4, x = <p(- <p~*x) 

 where it depends on the form of <p to have almost any algebraic 

 form we please for \J/. Thus if we assume 



_, a + b x 



<? X = — -, 



1 c + a x 



our form of \J/ will come out the same as (23), excluding the 

 arbitrary function. And if we consider that 



sin (a + b) = sin a . cos b + sin b . cos a 

 and 



cos sin" ' y = \/ 1 — y 2 , we have 



4, v x = <p j V 1 — <p x- . sin \- <p x . cos ( . . . .(2o) 



which, from the facility of giving finite and often rational values 

 to the transcendental parts, by choosing proper values for k, is 



