408 Professor Hamitton on Conjugate Functions, 
On a particular Class of Exponential and Logarithmic Function- Couples, connected 
with a particular Series of Integer Powers of Number-Couples. 
10. The theorem (69.) shows, that if we employ the symbols r,, ( a, a2) and F,, (4), b2) 
to denote concisely two number-couples, which depend in the following way on the 
couples (ai, a)) and (4, 2), 
a> 2) aoe Ay) Qo)” = 
F, Cas a) = d, iy ee : +6 Smo ihe eee ee (87.) 
25 (A, b,)' a(n by) (4, b,)™ 
Fe (6 6) = (1,0) +5 Teena gi Sr Yt pero SC a (88.) 
and if we denote in like manner by the symbol 
Pn (@, a.) ar (2, b.)) = Bn (4, ao by a, at: by) (89.) 
the couple which depends in the same way on the sum (4, @,) + (4,, 2), or on the 
couple (4, +2, ¢,+4,), and develope by the rule (69.) the powers of this latter sum, 
we shall have the relation 
{r,, (4, ay) X En (4, b,)% Fh (a; a2) + (G1, b2)) = 
(4, @)" Le ee CI ear eG ad t 
1 2 
1x2x3x_..m 2x3 xm 
(io 20 FGA et Gu BEND 
1x2x3x...(m—1) ISDN i tee linge acon cen 
+ 
(2%, %)' (5, &)” 
1 Ix 23K.) (90.) 
This expression may be farther developed, by the rule for the multiplication of a sun, 
into the sum of several terms or couples, (¢,, ¢:), of which the number is 
14+24+8+4...4m=0 (4D, (91.) 
and which are of the form 
t b- b,)* 
Cc : os (4, a.) ( ly 2 2. 
(, C1) = eeu eS Xa (92.) 
