2 ANNUAL KECORD OF SCIENCE AND INDUSTRY. 



when standing in need of food, was ordered by the oracle of 

 Delphi to construct an altar having double the volume of the 

 well-known cubical o-olden altar of that oracle. BoJimefs 

 Inaugural Dissertation, 1874. 



THEOREM IX THE EQUILIBRIUM OF FORCES. 



If we have to deduce the resultant of any number what- 

 ever of forces acting npon a rigid system of points, we may 

 begin by assuming for any given point, w, of the system two 

 new forces equivalent to any other of the given forces, and 

 which are parallel and equal to those, but have opposite di- 

 rections, and therefore hold each other in equilibrium. We 

 thus obtain successively for every pair of forces in the sys- 

 tem an equivalent in the shape of a simple force acting at n, 

 and a pair of forces w^hich can give rise only to rotatory mo- 

 tion. If now the whole system of forces is to be in equilib- 

 rium, it is necessary not only that the simple forces acting 

 at n shall be at equilibrium, but such equilibrium must also 

 exist among the systems of pairs of forces tending to give 

 rotatory motion. If equilibrium does not exist among the 

 forces, three cases may be conceived, of which the most gen- 

 eral is that in which both a motion of translation and a mo- 

 tion of rotation result from the application of these forces. 

 In this case it is possible to replace all the assumed element- 

 ary forces by two resultants, which will not act in the same 

 plane nor in the same direction. 



This subject has been handled in an interesting inaugural 

 dissertation by Dr. Dahmen, whence we derive the following 

 theorems : First, an indefinite number of pairs of forces may 

 be found, either of which will replace the original forces ; 

 and, secondly, if for each of these combinations of two in- 

 clined forces which are equivalent to each other, we connect 

 the ends of the lines representing them, thereby constructing 

 a tetrahedron, the solids so constructed will all have the same 

 volume. Inaugural Dissertatio7i of Dr. Dahmen^ 1874. 



THE VALUE OF THE DOZEN. 



As the derivation from the French douzaine implies, it is 

 generally presumed that a dozen means twelve things, but 

 in the Staffordshire potteries, and in the earthenware trade, 

 queen's-ware in Philadelphia, crockery in other places, a doz- 



