520 FOURTH REPORT — 1834. 



tional functions of pairs, enabling us to generalize any ordinary 

 algebraic equation from single quantities to pairs, and so to in- 

 terpret the research of all its roots, without introducing imagi- 

 naries. 



Without stopping to justify these definitions of sum and pro- 

 duct, which will probably be admitted without difficulty, Mr. 

 Hamilton inquires what analogous meaning should be attached 

 to an exponential pair, or to the notation {a, by-^'^^; or, finally, 

 what forms ought to be assigned to the conjugate functions 

 Us^, Vj.^, in the exponential equation 



(«, 6)('*> = {i,,,,j, v,,y) (e.) 



In the theory of quantities, the most fundamental properties 

 of the exponential function a^ =. (^ (x) are these : 



<f) {x) <f) (f ) = cf (ar + 0), and 4> (1) = a ; ... (f ) 



Mr. Hamilton thinks it right, therefore, in the theory of pairs, 

 to establish by definition the analogous properties, 



(«, bf^''J^ {a, 6)^^'") = («, bi' + ^y^''\ . . . (g.) 

 and 



(a, 6)^''«) = {a,b) (h). 



Combining these properties with the equation (e.) and with the 

 definition (d.) of product, and defining an equation between pairs 

 to involve two equations between quantities, Mr. Hamilton ob- 

 tains the following pair of ordinary functional equations, or 

 equations in differences, to be combined with the two equations 

 of conjugation : 



«x,y «|, n - 'V, y *'|, , = "x + |. y + r, 1 



and also the following pair of conditions, 



"1,0 = «> ^1,0=* (M 



Solving the pair of equations (i.), he finds 



u 



x,y 



=/(«' 2^ + /3' a:) . cos (a 3^ + ^ x). 



^'x,y =/(«' !/ + ^' x). sin{aci/ + ^ ar), 



a ^ a! ^' being any four constants, independent of x and y, and 

 the function/ being such that 



2 ,3 



/(r) = l +y+ j^2+j-^ + &c.; . . (ni.) 



