PARTICULARLY THOSE OF TWO VARIABLES. 
815 
26. The periodicity also easily follows. Since 6{x) and all its differentials are 
periodical in 4K and 6{y) and all its differentials in 4A, we have 
2KAlog r d? „ , , 
^>{£c+4m / K } 2/+4w , A}=e & my0^ K {x-\-4.m K)0 V}(t [y-\-ln A) 
= 6 -^me^)e hP {y) 
=<*>!#,«/}. 
(95) 
giving the two pairs of conjugate periods 4K and 0, 0 and 4A. 
27. If K' be defined by the relation 
~ n i = h SP 
then it is known that 
2 nix 
0* x(n +4«K') ^ K (x) 
Hence 
gKAlogr tf 8 2irea; 
{ x +4iK', y } = e~ K 0^ K (x) 9 Vt p (y) 
=p 4 
, x 4A , tf0„ p (y) , /4A, \n <»„*) 
=P ‘e K e ^ (x) ^ Jy) _^ logr ^ + l^ logr j - . . . 
_ 4 2 -^ 2KA log r 
^ 6 <fo 7T 2 
w r ^p(y) , w r VI *My) 
dy iri g g y 2! <?/ 
+p'% 
-4 Apg) 1 /2KA log ry 
dx* 2!\ 7r 
"^,,W_4A ,(y) , /1A, y 1 d%, p (y) 
iri » <fy» s / 2! <% 2 
2nix 
-pT^e~ K 
lo g r 
—- A log r 
tt2 dx 
dy 
and therefore 
<E> ja:4-4iK', y log r | 
2frix 
:p-4 e - K 
V? M/? O.A 2KA lo g r ^.aW d^, p (y) 
vw,<w- ^ dx dy 
/2KA logAn <P^,a(«) <«„ p M 
**~\ 7T 2 / 2! ^ dy* 
5 M 
MDCCCLXXXIL 
