1824.] Mr. Herapath on the Solution of^'x = x. 323 



which by introducing an arbitrary function, according to Mr. 

 Babbage's method, becomes 



v 



4/ x = ?"' . V<px (3) 



for the general solution of (1) whatever be the values of v and n. 

 When v = 1 , this expression gives 



i 

 lx = <p- l A" <$x (4) 



That excellent mathematician Mr. Herschel has given in Mr. 

 Babbage's 11th Prob. Phil. Trans, for 1815, a different expres- 



i 



sion for the value of 4* x ; namely, $ x = <p -1 { ( — 1)" <p x}; but 

 by what we have shown, this is the solution of 4>" x = — x not of 

 4>" x = x. 



Putting (3) under the form for the circular root of 1, we have 



^" x = <p \ (cos + V— 1 . sin ) <p xl (o) 



in which X is the semiperiphery to radius 1, and k is any integer. 



.-,, n + a ' n — a , . n + a 



And because cos A = cos x and sin A •— — 



n n n 



sin A, if we expound k by — — the double sign of (5) will 



occasion its values to circulate and to return for all magnitudes 

 of k into the same which take place between k = 1 and k = 



-or —7—, as n happens to be even or odd, or between a — or 



1 and a = n — 2, the increments of a being 2. It is also 

 evident if 11 be an integer, that the number of functional roots 

 will be n, and if n be a fraction, the number of roots will be equal 

 to the units in the numerator of this fraction ; so that if u be 

 irrational or imaginary, the number of functional roots will be 

 infinite. 



This solution being performed by the coefficient may be called 

 the coefficiential solution.' 



§ 2. Taking a x ■=. x h , we have 



a 1 x = x h \ 

 a 1 x = x h \ 



a" X — x k 



whatever be the value of n. Hence if b = 1, and introducing 

 the arbitrary function 



« 



fx = ?"' . tpx 1 " (6) 



v 2 



