526 FOURTH REPORT — 1834. 



nential notation from integral to fractional, to incommensurable, 

 to negative, and to imaginary quantities. He contends that 

 there are no propositions connected with the theory more fun- 

 damental than that, first, " in any exponential system, the ex- 

 ponent of the product of similar exponential functions of any 

 quantities is equal to the sum of the exponents of the factors" ; 

 and that, secondly, " an exponential function of 1 is equal to 

 the base." 



If a' = ij, the search of either symbol, (y the power, a the 

 base, X the logarithm,) as a function of the other two, fur- 

 nishes three principal problems. 



First, To find y in terms of a and x. 



The solution is a''=f{xf-^a) (2.) 



In this formula the notation /9 signifies cos 9 + -/— 1 sin 6, 

 cos 9 and sin 9 being functions of any real or imaginary quan- 

 tity 9, which, independently of 9 and 9\ fulfil the following con- 

 ditions : 



cos 9 cos 9' — sin 9 sin 9' = cos (9 + 9') "j 

 sin 9 cos 9' + cos 9 sin 9' = sin (9 + 9') > • • • (3.) 

 (cos 9)2 + (sin 9)2= 1 J 



Let a = r + -/^ s, r and * being real, then the notation 

 y ~ ^ a signifies 



2 i X + -4=^ cos" ' —r^ 1 + s/~^\ I f c, g - • (4.) 



In this formula i denotes 0, or any integer positive or nega- 

 tive- — ^ denotes 1 or — 1, according as s is not less or less 

 V' *2 



than 0. When s is positive or negative, s/ s^ denotes the po- 

 sitive square root of s^. The author makes considerable use 



s 

 of the class of expressions of which -^ is an example. They 



are extremely convenient in general formulae, particularly on 

 account of their property of obviating the necessity of separate 

 cases, cos" '9 represents the arc, when radius = 1, in the 

 first positive semicircle (including and tt) whose cosine = 9. 

 In the statement of propositions having limits, he suggests the 

 peculiar importance in these investigations of expressing clearly 

 whether the limits, or either of them, are to be taken inclu- 

 sively or exclusively. ^9 denotes the ordinary real Neperian 

 logarithm of 9. 



