324 Mr. Here/path on the Solution of $ n x = x. [Nov. 



which is another general solution of (1), and being obtained by 

 means of the exponent may be called the exponential. 



The exponent in this case being put under the same form as 

 the coefficient in the former case, admits the same observations 

 with respect to the functional roots, &c. We may combine 

 these two solutions, and have 



v 



V 



y x = p-' . 1" p x? (7) 



v 



In this solution the coefficient 1" may be, but is not necessa- 



r 



rily the same root as 1" the exponent ; that is, the indeterminate 

 integers k of the coefficient and exponent are not necessarily the 

 same. Hence the number of functional roots in this expression 

 is « 9 . For example, if n = 2, v as 1, and <p x — x, the number 

 of roots is four, 



This remark, therefore, destroys the opinion derived from the 

 analogy of algebraic equations, namely, that the functional equa- 

 tion 4-" x — x, n being a positive integer, has as many roots only 

 as n contains integers. It is indeed evident from the nature of 

 arbitrary functions, that the number of functional roots is inde- 

 finite, when the arbitrary function has its full scope ; but when 

 the arbitrary function is excluded, and not in any way antici- 

 pated, the number of functional roots is the same as the number 

 of algebraic roots of an equation of equal dimensions. In the 

 preceding instance the arbitrary function is in part anticipated 

 by the double solution ; and hence the reason that the number 

 of functional roots exceeds those denoted by the index. 



§ 3. If we set out with a function of the form 



a + bx 

 c + dx 



the 2d, 3d, 4th, &c. functions will evidently be of the same form. 

 And because a, b, c, d, are indefinite, any function of this form 

 may be conceived to be the 2d, 3d, or /th function of a like form ; 

 so that we may suppose 



, r it,- + b r x , , , a, + b, x 



V X = — an " ^ X = 



d r x c, + d, x 



whatever be the values of r and t. 

 Because 



,j/ ^' .r = 4/ ^' x = 4?+ ' x, 



it is manifestly immaterial whether in the value of ¥ x we sub- 

 stitute for x the value of tf x, or in the value of 4-' .r we substi- 

 tute for x the value of V x ; both results will be the same. 

 Making, therefore, these substitutions, and equating the corre- 

 sponding terms of the results, we obtain 



