328 Mr. Herapath on the Solution ofiTx — x. [Nov. 



This is another general expression for determining $• r x from 

 4-" i =x whatever be the value of v, r, or n, rational, irrational, or 

 imaginary. If we put r - v = 1, we easily deduce Mr. Horner's 

 expression for ^ x, namely, 



r~ a + b if x ~"\ 



2 k > 



, . J 4' - 2 b c cos - + c 2 



J/ .t = <p ~ ' 1 n 



T C <p j 



;a ( cos lT +1 ) 



(22) 



which was investigated for positive integral values only of n, but 

 which our general views show to be true for every value real or 

 even imaginary. 



Let us now apply these theorems to a few examples. Suppose 



first that n = 2, and k = 1 ; then cos — - = — 1, and the value 

 for d becomes ; + c — - which since the denominator vanishes 



« a. (^cos A + J ) 



must have b = — c. Differentiating the numerator and deno- 

 minator twice with respect to c and A respectively, we obtain d = 



2a8v , ^ denoting differentiation. Therefore, because J" c and o * 



are mutually independent, this value of d may be any thing, and 

 hence 



^..-f-'-j^ii; (23) 



4 ± rf as .i v / 



is the solution of <|> 9 x = r. It is rather curious that this solu- 

 tion is obtained on the hypothesis of k = | which Mr. Horner 



thinks cannot be, and obtained also from his own theorem. 

 Again, let n = 5 and k = I, then by Gauss's division of the 



circle cos -^ = V ~ , and therefore, 



n 4- J p a" 



*X = 



\~ 



2 6 2 - ic (v' 5-1) + 2c 2 



— $ x 



«(|/ 5 + 3) 



which is the first and only solution I have seen of vj/ 5 x = x. 



I shall not for brevity's sake stop to compute other cases of 

 integral functions, but shall just give an instance, the first, I 

 believe, that has been given, of the solution of a fractional func- 



O Q J. \ 8 A 



tion. Suppose n = - and k = 1, then cos = cos— = — 



- , and, therefore, (22) 



