223 
tions Mf,, Vf,: and this arises from the mixture of real and 
imaginary quantities in them. 
«8, The solution just mentioned, viz., #,+ F,, might be 
written in the form 
{1+D,? (D? + D3)}> (fit ahr). 
This transformation suggests an elementary process, by means 
of which the solution of Laplace’s function, in the form ofa 
series arranged according to ascending powers of z, may be 
obtained without recourse to imaginaries. Let the equation, 
(Di + D} + D?) V=0, 
be integrated twice with respect to x; ¢, and ¢,, two arbitrary 
functions of y and 2, being successively introduced in the in- 
tegration ; it will then assume the form 
{1+.D,? (D? + D3)} V=a¢2 + or 
Hence we shall have, 
V=(1+D,? (Dj + D3)}7 (#2 + di): 
The development of the operations here indicated will actually 
produce a result equivalent to Lagrange’s. So long ago as in 
February, 1848, I had suggested this mode of treating diffe- 
rential equations; but I had then little notion of the possibi- 
lity of applying it with any success in the case of an equation 
so intractable as that of Laplace’s coefficients.” 
Dr. Todd presented a rubbing made by him from an in- 
scribed tombstone in the north transept of the church of Gal- 
way. It bears the following inscription :— 
- HIR - LIETH - THE - BODI- OF - ON - MORIZRTAH - OTIER- 
NAGH - AND - HIS: WIF - KATERINA - NIGONOHW - AND - HIS - 
BROTHER - TEIGE - OG- CVPERS - AN°: DNI: 1580 - 
The stone is elaborately ornamented, and bears on it also 
a representation of an adze and square, or rule, the emblems 
of the trade of coopers, to which the brothers O’Tiernagh be- 
longed. 
