MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 525 



that is to say, the function of the sum of two arguments is the product of the 

 functions of those arguments separately. This is merely a statement, in modern 

 notation, of the idea which guided Nepair to the invention of logarithms. 

 Extending the above equation, we find 



<p (t + io + v) = <pt . <pu . <pv, 



which, on supposing t, u, v to be all alike, gives in general 



9 (nt) =( 9 t)» . 



If, according to the usual practice, we put e for the value which this function 

 assumes when the primary is unit, that is, if we put 



e = l + l + + iTO + OXI+ etc " 

 we easily obtain for every integer value of n 



^ = e =1 + l + + TX3 + etc - 



Also, if we put nt = 1, or t = - , 



n 



f l = e = ? (-J , whence <p (-A = e n 



so that the above development of e n is true also of fractional values of n, wherefore 



t t 2 t s 

 ^ = e * =1 + i + l72 + TX3 + etc - 



Thus the exponential function is reached in the first step of our researches 

 into the theory of recurring functions. 



The properties of this remarkable function are too well known to be in need 

 of any elucidation here. I may, however, indicate a rapid approximation to the 



n th root of e, obtained by help of Brounckee's continued fractions. On expand- 



i 



ing the series for e n into a chain-fraction, we obtain the successive quotients 

 1, n — \, 1, 1, 3n — 1, 1, 1, 5n — l, 1, 1, 7n — l, 1, etc. ; and on 

 computing from these the series of converging fractions, we find that the 1st, 

 4th, 7th, 10th, etc., of them form the following progression : — 



2n 6n lOn 14n 



1 1 2n + l 12n 2 + 6rc + 1 12(k 3 + 60n 2 + 12w + 1 

 - 1 1 2n - .1 12n 2 - 6n + 1 12(k 3 - 6(k 2 + 12n - 1 ' 



in which the successive multipliers 2n, 6n, lOn, etc., form an arithmetical pro- 

 gression, of which the common difference is 4n. 



6. The converse of the exponential function is the logarithmic ; thus, while e' 

 is called the exponential function of t, t itself is called the logarithm of e\ If we 



