327 



These are the fundamental equations for the comparison 

 of hyperbolic sines and cosines. 



2. The distributive symbols -|-'» +*? +*? + i enable us 

 to construct the ordinary double algebra. The equation 

 +M + 1 = makes -f-* 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 



+ * 1+^1 = 0, 



which is a consequence of 



+ M + I =0. 



3. With the distributive symbols +', -|-% +, 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. Starting with the primary symbol -{-' we might frame 

 an algebra with two imaginaries, viz., -|-^ and -1-^ and three 

 reals, -f-^, +% +, related to one another by the condition 



-f* 1 -l-M 4- 1 = 0. 



1 

 Developing e"^' '^ we find it equal to 



1 2 



A + U -j- V , 



X , ju , and V representing series in which the signs of the 



terms are successively -H , +% +% -l-> +% +*> &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, and of 

 2 E 2 



