OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
531 
From (1-18), we have 
Q-H> (cosh xjf) = f - 7 = 
Jii \/ - 
e~P‘“ 
V 2 cosh % — 2 cosh -vp 
c/?<. 
Hence 
Q_i +pt (cosh i//) + Q J pi (cosh i/r) = 2 [ 
J ; 
COS 
^ \/ 2 cosh % — 2 cosh \fr 
du, 
hence defining K/ ( — cosh xjj) by means of the formula 
K,, (— cosh i/») = — cosh 2 ?n 
7 r 
cos pu 
du, 
we 
have 
COS/>7T 
7T 
0 \/2 cosh u — 2 cosh -v/r 
{Q_i +Pl (cosh i/>) + Q_i_ pt (cosh i/,)} = K y , (— ccsh i/»). 
It thus appears that the known expressions for these functions are ii 
derivabie from the general formulae obtained in the present memoir. 
3 Y 2 
mediately 
