MATHEMATICS AND ASTRONOMY. 
51 
that is 
?( Aa ^ a = cosh A -f- a% sinh A ’ a 2 ; 
The expression a A , when no period is expressed, is understood to have 
the period £; in other words the area"^ is hounded by a , circular arc. 
Let ha A denote the same when the bounding arc is the equilateral hyper 
bola (fig. 10). Then the rectangular components OM and MQ of the hy 
perbolic versor which has the axis a and the area ^ are commonly de 
noted by cosh A and sink A, so that 
7T 
ha A = cosh A + sinh A • a 2 
gether with the circular versor sinh A - a 2 . 
, ^ 
he 
ha A = cosh A + sinh A * a 2 , 
_ A , Aa 2 
To prove that ha —he 
W e have 
= 1+— +— + 
~ 2! ‘ 41 ~ 
A 3 
+ + it + ) ' a ' 2 
This is an essentially different expansion from the circular. It may be 
7T 7T 
A ^ A ”2" — TV 
denoted by h e , and it differs from that for e Aa in having a 2 u? — i. 
Similarly 
A -y 
ha — cosh A — sinh A • a 2 , 
= h e" 
-Aa^ 
rr, , A Aa 7r 
To compare ha with e 
e AaU = cosh A + sinh A • a 
n J7T 
= cosh A + a 2 sinh A - a 2 ; 
TT 7T_