4 ANNUAL KECORD OF SCIENCE AND INDUSTRY. 



works contain in the greatest completeness, and in a well- 

 arranged series, the principal results of some twelve years 

 of laborious computation wiiich the author had undertaken 

 in order to lay a foundation for the analysis of the prime 

 numbers by deriving them from the roots of unity on the 

 basis of Kummer's theory. In the study of the theory of 

 the complex numbers, the computations are so exceedingly 

 difficult and laborious that it is scarcely possible for indi- 

 vidual investigators to provide themselves in any special 

 case with the necessary material of observation in order to 

 find therein some base for theoretical investigation. There- 

 fore the computations of Professor Reuschel have frequently 

 proved to be valuable preliminary labors, and bring praise 

 to the perseverance and devotion with which he has for so 

 many years labored on his work. Unfortunately, the acci- 

 dental death of Reuschel, on the 22d of May of this year, in 

 the sixty-third year of his life, deprives him of the pleasure 

 of reaping that benefit from his work that should naturally 

 have been his. Monatsbericht k. Akad. der Wiss.^ BerUiiy 

 1875,238. 



HYPERBOLIC FUNCTIONS. 



In pure mathematics, we notice the publication, in the 

 Memoirs of the Scientific Society of Bordeaux, of an essay 

 by Loisant on hyperbolic functions, in which he especially 

 investigates the correlation between the co-ordinates of any 

 point of an ellipse and any point of the hyperbola having 

 the same axis. His investigations, which seem to have been 

 carried out under the advice of Realis, lead apparently to 

 numerous curious and novel mathematical relations suscep- 

 tible of being practically useful in very many problems of 

 applied mathematics. J/em. So. Soc. of Bordeaux^ 1874, 101. 



CUEIOUS CURVES AND EQUATIONS. 



In a memoir on a certain class of transcendental curves, 

 Professor H. A. Newton, of Yale College, states that alge- 

 braic curves have hitherto been studied more than transcend- 

 ental. A few of the latter have been given in the text- 

 books, but attempts to classify the numerous varieties of 

 transcendental curves have been rare. From the form of a 

 transcendental curve it is not easy to state an equation that 



