528 FOURTH REPORT — 1834. 



is equal to any quantity furnished by formula (5.), and con- 

 versely, if any value of a' = y, a cannot but be equal to 

 S07ne one of those quantities. 



If, however, the problem be to find a, such that the i'** value 

 of fl-^ may = y, i being given as well as x and y, it may be 

 impossible to solve the problem by any value of a represent- 

 able by admitted algebraic symbols, or reducible to the form 

 r + \^ — I s. The general result of the author's investigations 

 on this branch of the subject is, that being given the equation 



a,- ^ * =^ + V' — 1 5^, we shall have, when r is not = 0, 

 at least n algebraically representable solutions, and may have 

 » + I solutions, if r^+ s^ be greater than m a/ r^ ; and that we 

 can have at most but one such solution, and may have not even 

 one, when r^ + s'^ is not greater than -v/ r'^. When \/ r^ is 

 equal to or greater than 1, one representable a at least may 

 always be found to satisfy the equation. The " chance of re- 

 presentability" of a, when a.'^ v - 1 »• jg given, when i is for 

 the first time taken at random, and r is not = 0, may be de- 



noted by ,~ , certainty being denoted by 1. Let r be = 0, 



then the equation becomes a/ ~ * =p + \/—lg. This equa- 

 tion, i being given, as well as s, p, and q, will have an infi- 

 nite number of roots for a, if — I —-===== be greater than 



* ^p^ + q^ 

 {2i — 1) 'jr, and not greater than (2 ^ + 1 ) w ; otherwise, it will 

 have not one representable root. 



Third, The third problem is to represent all the logarithms 

 of a given quantity in a given base. 



Let ft*' = y, then every quantity which, being substituted for 

 .V, allows any value of «''', as explained by (1.), to be equal to y, 

 is, according to the author's definition, a " logarithm" of y 

 in the base a. 



The solution of this third problem is 



-=pl («•' 



Any particular logarithm (x) will be of the form '2 ^ , z and i 



being arbitrary independent integers. 



i, in the denominator of the preceding formula, names, ac- 

 cording to the author's nomenclature, the "order" of the lo- 

 garithm, and t, in the numerator, its " rank" in that order. 



