3 
We may, however, make most advantageous use of the 
equation (2) by assigning arbitrary forms to the functions 
and ~ contained in it, and so construct soluble forms ad libi- 
tum. 
When the equation (1) wants the term involving Dy, we 
have 
¢+W=O0and gf + yp = B. : 
Hence —¢?-g=B. (4) 
Now, as the second term of the equation (1) can always be 
banished by a change of the dependent variable, we have ar- 
rived at the remarkable result that the solution of the general 
linear differential equation of the second order depends upon 
that of the equation (4), whose form is particular and un- 
changing: and this result is practically important; for if we 
tabulate the values of ¢? + ¢’ for all values of ¢, we should have 
the solutions themselves of linear differential equations of the 
second order tabulated at the same time. 
By interchanging the symbols x and D in the preceding 
formule, according to the method pointed out by Dr. Har- 
greaye, we are led to a series of general and interesting results, 
Dr. Todd made some remarks on the fresco painting in 
the Abbey of Knockmoy, in the county Galway, of which a 
fac-simile copy, the exact size of the original, was exhibited in 
the Antiquarian Court of the Dublin Exhibition. 
The public are indebted for the preservation and exhibi- 
tion of this ancient monument of Irish art to the zeal of Dr. 
John Lentaigne, at whose instance, and by whose personal 
exertions, the fac-simile was obtained for the Committee of the 
Exhibition. The following account of the manner in which 
the inscriptions were deciphered is given in a letter dated 13th 
June, 1853, addressed to Dr. Todd by Mr. Eugene Curry : 
“John Lentaigne, Esq., on the part of the Committee of 
the Great Industrial Exhibition, having done me the honour 
to request me to accompany him to the ruins of the once 
B2 
