4 ANNUAL RECORD OF SCIENCE AND INDUSTRY. 



through the discrepancies of clocks and watches is very con- 

 siderable, and is directly felt by each individual in the miss- 

 ing of appointments or the needless loss of time in waiting. 

 On very many accounts the country throughout the whole 

 region east of the Rocky Mountains would be benefited by 

 the introduction of some uniform standard of time which 

 should replace the innumerable and often erroneous "local 

 times," and by which not only railroad, telegraph, and stock 

 business might be managed, but which should be adopted 

 also in governmental and in private matters. Description 

 of the City Hall, Pittsburgh. 



PROPERTIES OF PRIME NUMBERS. 



As the conclusion of an investigation by Goering into the 

 "Theta" functions of Jacobi, and as an application of his re- 

 sults, the author shows that every prime number of the form 

 6m-f-l is always divisible, although only in one special way, 

 into the sum of a simple and a triple square ; and, again, 

 that the product of n prime numbers of the form 6m -J- 1 can 

 always be considered as the sum of a simple and a triple 

 square. Goering, Inaugural Dissertation, 1874, p. 382. 



APPLICATIONS OF PEAUCELLIER CELLS. 



Mr. Darwin has given an account of some applications of 

 what are now familiarly known as Peaucellier cells. Among 

 other things he illustrates the fact that it might become pos- 

 sible to construct by means of these a model that shall give 

 an ocular and correct proof of the elliptic motion of the plan- 

 ets about the sun, under the influence of the force varying in- 

 versely as the square of the distance in that fixed point. Mr. 

 Sylvester states that he himself had attempted the same 

 problem, but failed. 



Hamilton's equation of motion. 



A decided advance in the principles of theoretical mechan- 

 ics seems to have been made by Professor Miiller, of Zurich, 

 who has developed certain considerations based upon what 

 is known as Sir William Hamilton's general equation of mo- 

 tion. That distinguished mathematician has shown that 

 when a system of material points moves under the influence 

 of forces proceeding from the reciprocal attraction and re- 



