390 
dp = rdz + sdy | 
dq = sdx + tdy { 
and integrate again. The values of p and qg thus found are to be substi- 
tuted, finally, in the equation 
dz = pdx + qdy, 
and by a third integration the solution required is, in general, determined. 
It willbe evident that in the processes of successive integration indicated, 
five arbitrary constants are introduced. 
JosepH Husanp Smuitu, Esq., read a paper— 
ON THE ANCIENT NORSE AND DANISH GEOGRAPHY OF IRELAND. 
AFTER some observations upon communications made to the Academy by 
the late Mr. George Downes, in the years 1838-9 and 1841, and having 
referred also to the labours of Dr. William West, who had died in 1887, 
while engagedin preparing an Essay on the Ancient Geography of Gauland 
the British Isles, intended to be laid before the meeting of the British As- 
sociation, then approaching, Mr. Smith proceeded to read some passages 
from recently-discovered copies of our ancient annals, illustrative of the 
military occupation of Ireland by the Vikings, who arrived from Seandi- 
navia at various periods, commencing towards the close of the eighth cen- 
tury of our era, and continued a long series of contests with the abori- 
ginal inhabitants of Ireland, and subsequently with each other, during 
the four succeeding centuries. 
From passages in the Icelandic and other Sagas, he showed that a 
constant intercourse had been maintained at first by the Norwegian sea 
rovers, who had gained a footing in the Orkneys, and afterwards by 
others, described in the Irish annals by the names of Lochlanns, Gen- 
tiles, and Finn Ghaoill, and Dubh Ghaoill, not only of a warlike nature, 
but also of a trading or mercantile character. The names of various 
localities which he instanced,—beginning with the Orkneys, the Faro- 
Islands, and the Hebrides,—all, he contended, indicated the progress 
of the war vessels of the various tribes; and he pointed out the harbours 
and islands along the eastern coast of Ireland, whose names (some as re- 
corded in the Sagas, and others preserved to the present day) were un- 
mistakeably of Norwegian and Icelandic origin, and quite distinct from 
the Celtic names by which these places were known to the Irish writers of 
the same period. Among others, Mr. Smith instanced Strangford bay, 
Carlingford, the little islands of Lamb-ey, and Ireland’s-ey; the harbour 
of Bullock or Blowyck, and the adjacent island and sound of Dalkey ; the 
headlands of Wicklow, or Wykynge-lo, and Arc-lo; the harbours of 
Wexford and Waterford, as well as that of Smerwick, on the western 
coast, not far from the River Shannon,—in all of which the Norwegian 
or Danish Vikings had sheltered successive fleets, and in most cases 
erected on their shores military fortresses of considerable strength. Some 
other places named in the Sagas,—for example, Gunvallsborg,—Mr. 
Smith admitted he was as yet unable to identify satisfactorily with any 
modern localities. He then adverted to the modern names of three of 
