. 
. 
223 
tions Mf, Mf,: and this arises from the mixture of real and 
imaginary quantities in them. 
«8, The solution just mentioned, viz. F,+ F, might be 
written in the form 
(1+.D,? (Di + Di)}> (fit afr). 
This transformation suggests an elementary process, by means 
of which the solution of Laplace’s function, in the form of a 
series arranged according to ascending powers of 2, may be 
obtained without recourse to imaginaries. Let the equation, 
(Di + _D} + D2) V=0, 
be integrated twice with respect to 2; $, and g,, two arbitrary 
functions of y and z, being successively introduced in the in- 
tegration ; it will then assume the form 
{ 1 3h i Oh (D; ae D3)} V=29, ate pi. 
_ Hence we shall have, 
V=({14+ Dy? (D} + D3)}7 (#24 o:)- 
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 
 arepresentation of an adze and square, or rule, the emblems 
_ of the trade of coopers, to which the brothers O’Tiernagh be- 
longed. 
b) 
