* 
. 
327 
These are the fundamental equations for the comparison 
of hyperbolic sines and cosines. 
2. The distributive symbols +*, +4, +3, +, enable us 
to construct the ordinary double algebra. The equation 
+!141=0 makes +! real as well as +. But +? and +' 
are both imaginary, as compared with + and +}, though 
they admit of being compared together because of their rela- 
tion 
4-71 441 = 0, 
which is a consequence of 
+414+1=0. 
3. With the distributive symbols +*, +3, +, supposed 
to be heterogeneous, we might construct an algebra of one 
real and two imaginary symbols. This algebra would be vir- 
tually equivalent to the triple system discussed by Mr. Graves 
in former communications to the Academy. 
4, Statting with the primary symbol +5 we might frame 
an algebra with two imaginaries, viz., +9 and +5, and three 
reals, +", +3, +, related to one another by the condition 
+)514+7141=0. 
Developing ot? we find it equal to 
b ; 
2 Se ee 
r o hy and y, representing series in which the signs of the 
_ terms are successively +, +), +5, 4, +4, +4 &c. Between 
these series and the ordinary trigonometric series, expressing 
the sine and cosine in terms of the arc, there exist many 
remarkable analogies. 
_ Mr. J. J. A. Worsaae, of Copenhagen, in continuation of 
_ his former communication to the Academy, gave a review of the 
_ different descriptions of Danish and Irish antiquities, ani of 
2°52 
