4.20 Professor Haminton on Conjugate Functions, 
power- couple, and in which the other factor may be calculated by the following ex- 
pression, 
(1, 0) (ts 2)» (Cay, ay) x (0, 2 w 7) 
=F(-207 &, 2w7 2X) 
=e —2em (cos Quart, sin 2Qwr%). (170.) 
For example, 
(1, 0) (t» %)— (1, 0), (171.) 
and 
(6, 0) (#1 7) = (ERD) R (172.) 
also 
(¢, 0) (i %) p(w, a) x (1, 207) (173.) 
On Exponential and Logarithmic Function- Couples in general. 
16. It is easy now to discover this general expression for an exponential function- 
couple : 
® (2, 2) =F (CX, X2) x (Gis G2) 5 (174) 
in which (a,, a2) is any constant couple, independent of (a, a). This general expo- 
nential function © includes the particular function r, and satisfies (as it ought) the 
condition of the form (128.), 
® (21, a) ®D (YM, Yr) =P (a, +s Hop) +Yo) 5 (175.) 
its base, or base-couple, which may be denoted for conciseness by (4, 52), is, by the 
11th article, the couple 
(Gib) =o Oe 0) "Gi a3); (176.) 
and if we determine that integer number w which satisfies the conditions 
Q,-2w7 > —7, —L2wT PT, (177.) 
we shall have the general transformation 
® (@y 24) = (by by) Fo ™), (178.) 
