982 
which is not a point of the Net, can be included within a Geodetic Triangle 
on that surface, of which the corners are net-points, and of which the sides 
can be made as small as we may desire.” 
In fact, the surface is a rational one [129.], or the coefficients of its 
equation may be made whole numbers; and therefore every rational 
line [8.], from any one net point, or rational point, upon it, if not hap- 
pening to ¢ouch the surface, is easily proved to meet it again, in another 
rational point: whence, with the aid of a lately mentioned principle 
[127.], the theorem evidently follows. 
Reap, a letter from G. V. Du Noyer, Esq., dated Arklow, April-20, 
1861, inclosing some drawings of antiquities, and the following notes in 
explanation :— 
“As a contribution to my portfolio of drawings in the Royal Irish 
Academy Library, I send the accompanying sketch of a: very perfect 
Ogham-bearing pillar-stone, now lying on the road-side, close to and 
north of the old church of Castletimon, in the parish of Dunganstown, 
county of Wicklow, and at a distance of eight miles to the north of 
Arklow. 
“‘This Pillar, which is a well-smoothed block of crystalline green- 
stone, measuring 4 feet 10 inches in length, is called ‘The Longstone,’ 
and is held in much veneration by the people of the neighbourhood. 
“In a field to the south of Castletimon old church, and at the dis- 
tance of four hundred yards south of the Ogham stone, are the ruins of 
what was once a magnificent cromlech, the covering stone of which mea- 
sures nearly 11 feet square, by 2 to 24 feet in thickness, being, like the 
pillar, formed from a block of greenstone. Of this I also send the Aca- 
demy a sketch, with a rough plan of the stones forming it, to show their 
original relative position. In the Ordnance Survey Map, No. 36, Co. 
Wicklow, these relics are erroneously grouped together, and placed on 
the road-side to the north of the old church, mention being made of them 
as ‘Cromlech in ruins, called the Longstone.’”’ 
