122 
shall forfeit the sum of one hundred pounds sterling ;’”—whereupon it 
was— 
Resolved—‘ That the act of the President and Council in accept- 
ing this loan of antiquities from the Royal Dublin Society be ratified 
by the Academy, according to the terms of the receipt signed by the 
President.” E 
The President having ruled that the vote of adjournment of the 
Academy at its last meeting had the effect only of postponing the dis- 
cussion of the recommendation of the Council passed at their meeting 
of 7th February, 1859, viz. :—‘‘ That a subscription be opened for the 
purpose of completing the Catalogue of the Museum,’’—the discussion 
was resumed accordingly, and on a division the proposition of the 
Council was adopted by a majority of four, the numbers being—15 for 
the Resolution, and 11 against it. 
The Secretary of the Academy read_a letter from W. R. Wilde, Esq., 
on the subject of the preparation of the Catalogue ;—whereupon it was— 
Moved by J. F. Waller, LL. D., and seconded by the Rey. Charles 
Graves, and resolved:—‘‘ That Mr. Wilde’s letter be referred to the 
Council, with a view to its being entered on the Minutes of the Aca- 
demy.”’ 
Sre Wirr1am Rowan Hamitton, LL. D., communicated the follow- 
ing paper :— 
ON SOME QUATERNION EQUATIONS CONNECTED WITH FRESNEL’S WAVE-SUR- 
FACE FOR BIAXAL CRYSTALS. 
1. Tur ellipsoid of which the three semi-axes are usually denoted as 
a, b, ec, in statements of the Fresnelian theory of the wave-surface in a 
biaxal crystal, being here represented by the equation, 
Spe =1, 
where the vector function ¢ has the distributive and other properties 
described by Sir W. R. H., in his Seventh Lecture on Quaternions, it 
follows from the physical principles, or hypotheses, of Fresnel, that a 
small displacement, ép, of a molecule of the ether in a crystal, gives rise 
to an elastic force, which may be denoted by $1 ép. But if this dis- 
placement, ép, be (as is assumed) tangential to a wave-front in the 
medium, to which the vector » is normal, and of which the tensor Zu 
denotes the slowness of propagation, so that ~ may be called the InpEx- 
Vector, then the tangential component of the elastic force must admit 
of being represented by w®ép. Hence the normal component of the 
same force (supposed by Fresnel to be destroyed by the incompressi- 
bility of the ether) must admit of being denoted by the symbol, 
(f'— 1°) ép; 
which symbol must, therefore, admit of being equated to a vector of the 
form p'ém, 6m being a small scalar. We are, therefore, at liberty to 
write the following symbolical expression for the displacement supposed 
by Fresnel to exist : 
bp =(p'— #7)" wo enn. 
