\ 



TRANSACTIONS OF THE SECTIONS. 521 



and having established the following, among many other gene- 

 ral properties of conjugate functions, that if two such functions 

 be put under the forms 



«'x,y=/(Px,y)-C0s9,,yl 



"x,y=/0'..y)-sin9^,yj * 



y* still retaining its late meaning, the functions p^^ ^^y are also 

 conjugate, he concludes that the 4 constants of (1.) are con- 

 nected by these two relations, 



/3' = + «, a' = - /3, (o.) 



so that the general expressions for two conjugate exponential 

 functions are : 



^x.y =/(« X - ^ y) . cos {a y + ^ x)A 



f • • • (P-) 

 ^x,y =/(* X - ^ y) . sm {a. y + ^ x);\ 



and it only remains to introduce the constants of the base-pair 

 (a, b), by the conditions (k.). Those conditions give 



a =/(a) . cos /3, b =f{a) . sin /3, (q.) 



and therefore, finally, 



r' y (r.) 



/3 = ^o + -'«'r, J 



i being an arbitrary integer, and ^q being a qviantity which may 

 be assumed as > — tt, but not > tt, and may then be deter- 

 mined by the equations 



cos Bq = , sin /Sq = — . . . (s.) 



" Va^' + b^ " Va^ + b^ 



The form of the direct exponential pair (a, 6)(^'^), (or of the 

 direct conjugate exponential functions u, v,) is now entirely de- 

 termined ; but the process has introduced one arbitrary inte- 

 ger i. 



Another arbitrary integer is introduced by reversing the 

 process, and seeking the inverse exponential or logarithmic 

 pair, 



{x, y) = Log. {u,v) (t.) 



{a,b) 



=/ 



