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1. Its velocity will vary inversely as the sine of the per- 
pendicular are let fall from s on the great circle which isa tan- 
gent at P to the trajectory described by the point. 
2. The force in the direction of the are sp is equal to 
2 
ae inti being the velocity of the point, y the radius of the 
Y 
osculating circle, and @ the angle between sp and the tangent 
are. 
The proposition may be readily proved by means of these 
principles, taken in conjunction with the following property 
of spherical conics : 
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 are 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 
