TRANSACTIONS OK THE SECT-IONS. 519 



On Conjugate Functions, or Algebraic Couples, as tending to 

 illustrate generally the Doctrine of Imaginary Quantities, 

 and as confirming the Results of Mr. Graves respecting the 

 Existence of Two independent Integers in the complete ex- 

 pression of an Imaginary Logarithm. By W. R. HAMiLr 

 TON, M.R.I.A., Astronomer Royal for Ireland. 



Admitting, at first, the usual things about imaginaries, let 



u + v \/ — I = <^.{x + y^/~^[), . . , . . (a.) 

 in which x, y are one pair of real quantities, and u, v are an- 

 other pair, depending on the former, and therefore capable of 

 being thus denoted, w^,j„ v^^y. It" is easy to prove that these 

 two functions, w^,y, v^^, must satisfy the two following equation^ 

 between their partial differential coefficients of the first order : 



du _dv du _ dv 



dx~ dy' dy~ dx ^ '' 



Professor Hamilton calls these the Equations of Conjugation, 

 between the functions u, v, because they are the necessary and 

 sufficien t con ditions in order that the imaginary expression 

 u -{- V a/ — I should be a function o^ x -{■ y »/ — 1. And he 

 thinks that without any introduction of imaginary symbols, the 

 two real relations (b.), between two real functions, might have 

 been suggested by analogies of algebra, as constituting be- 

 tween those two functions a connexion useful to study, and as 

 leading to the same results which are usually obtained by ima- 

 ginaries. Dismissing, therefore, for the present, the concep- 

 tion and language of imaginaries, Mr. Hamilton proposes to 

 consider a few properties of such Conjugate Functions, or Al- 

 gebraic Couples ; defining two functions to be conjugate when 

 they satisfy the two equations of conjugation, and calling, un- 

 der the same circumstances, the pair or couple {u, v) a function 

 of the pair {x, y). 



An easy extension of this view leads to the consideration of 

 relations between several pairs, and generally to reasonings 

 and operations upon pairs analogous to reasonings and opera- 

 tions on single quantities. For all such reasonings it is neces- 

 sary to establish definitions : the following definitions of sum 

 and product of pairs appear to Mr. Hamilton natural : 



{x, y) -f- {a, b) =. [x + a, y -^ b), (c.) 



{x, y) X (a, b) = {x a — y b, X b + y a), . . . (d.) 

 and conduct to meanings of all integer powers and other ra- 



