163 
MONDAY, MAY 9, 1859. 
James Hentuorn Topp, D.D., President, in the Chair. 
Siz W. R. Hamitron resumed the reading of his paper— 
ON CERTAIN EQUATIONS IN QUATERNIONS, CONNECTED WITH THE THEORY OF 
FRESNEL’S WAVE SURFACE. 
Ir Spdp=1 be the equation of an ellipsord (or, indeed, of any other central 
surface of the second order), then the ¢dentity, 
PV ede =p - p'=(-p”)p, 
proves that the vector, «= p—p", is perpendicular at once to p and to 
Vpbp. But Vp¢ has the direction ofa line tangent to the surface, which 
is also perpendicular to the semidiameter p, because 9p has the direction 
of the normal to the surface, at the end of that semidiameter. Hence o 
is normal to the plane of the section, whereof p is (not merely a semidia- 
meter, but) a semiaxis ; the other semiaxis having the direction of Vpdp. 
But p=(?-p”*)'o; pand Lo; 
“. 0= Sa (9-p-?)16; (1) 
and this last formula, which (when developed either by the aBy or by 
the «« form of @), is found to lead to a quadratic equation, relatively to 
p’ (or to Tp”), must, therefore, give, in general, the ¢wo scalar values of 
the square of a semiaxis of the section perpendicular to ¢, when the direc- 
tion of this normal oa, or of the plane itself, is given. 
Suppose now that the normal g is erected at the centre of the ellipsoid, 
and that its length is made equal to the length of one of the semiaxes p 
of the section, we shall have, of course, Tc=Tp, and may write 
0=So(9- o*\ 1, (2) 
as the equation of the Jocus of the extremity of o: that is, according to 
Fresnel, of the wave surface. But this is just the form (b), when we 
write p for o. 
Rey. Professor Graves read a paper—‘‘On a Gaulish Inscription,”’ 
noticed by M. Pictet in his recent work. : 
Mr. Wilde read the first part of his paper—“ On the unmanufactured 
Animal Remains in the Academy’s Museum.” 
Rey. Joseph A. Galbraith read a note by Lieut. Renny, explaining 
that the table printed at p. 639 of his paper, read June 14, 1858, was 
founded upon figures published in the monthly tables of the “ Biblio- 
theque Universelle ;” and that it now appears, from a communication 
by M. Plantamour, that a correction of + 0°79 should be applied to these 
numbers, as the equation of the Geneva barometer. 
