310 



1. Its velocity will vary inversely as the sine of the per- 

 pendicular arc let fall from s on the great circle which is a tan- 

 gent at p to the trajectory described by the point. 



2. The force in the direction of the arc sp is equal to 



r-^, V being the velocity of the point, y the radius of the 



osculating circle, and d the angle between sp and the tangent 

 arc. 



The proposition may be readily proved by means of these 

 principles, taken in conjunction with the following property 

 of spherical conies : 



A tangent arc being drawn at any point on a spherical 

 conic, if a perpendicular be let fall upon it from a fixed point, 

 and if a second perpendicular be let fall from the point of con- 

 tact on the polar of the fixed point, the quotient of the sines 

 of these two perpendiculars will always be proportional to the 

 tangent of the normal arc at the point of contact. 



This very general theorem is its own polar reciprocal. 



Mr. J. J. A. Worsaae, of Copenhagen, being requested to 

 give an account of the formation of the Museum of Antiqui- 

 ties in that city, made a communication to the following effect : 



" It is a very well known fact, that but few countries in 

 the north of Europe escaped invasion or conquest by the Ro- 

 mans. Among those few, however, Ireland and Denmark 

 are specially to be named ; and on that account it is certainly 

 more than a mere accident that these two countries are in 

 possession of some of the best collections of national antiqui- 

 ties in Europe. I have had the opportunity of repeatedly in- 

 specting the very interesting collection of the Academy, and it 

 has been told me, that the comparatively large number of Irish 

 antiquities there assembled has been brought together in a 

 short time, but under circumstances of considerable difficulty. 

 Our collection of national antiquities in Denmark has like- 

 wise been founded under great disadvantages ; and perhaps it 



