248 report— 1859. 



The term in P,, treated in like manner, gives '113 and "116 as the values of \ /—. 



The true value is probably about '117, ai 

 this by an obvious arithmetical process. 



The true value is probably about -117, and the value of the ratio - can be found from 



c 



On an Application of Quaternions to the Geometry of 'Fresnel's Wave-surface. 

 By Sir William Rowan Hamilton, LL.D. fyc. 

 Abstract of Formula, 

 p = vector of ray-velocity ; p. = index-vector, or vector of wave-slowness ; 

 S/*p= — 1, Sp8p=0, Spo>=0 (equations of reciprocity); S/> = vector of displace- 

 ment, or of vibration ; (p~ 1 8p = vector of elasticity, or of total resulting force of 

 restitution ((p being the same symbol of operation as in the Seventh Lecture on 

 Quaternions, by the present author) ; p~ 2 Bp = a vector, representing the tangential 

 component of elasticity ; .". (<£ _1 — p.~ 2 )8p=norma\ component of elasticity =p.~ 1 &m, 

 8m being a scalar ; .'. dp = (<p~ l —p~ 2 )~ i p~ 1 dm, and Sp.~ 1 dp = ; .-. the formula, 



O=S f t al (0- 1 - / *- 3 rV- 1 , (a) 



is a symbolical form of the equation of the index-surface, or of the surface of wave- 

 slowness, to which the wave itself is reciprocal. Hence, by the equations of reci- 

 procity given above, or simply by changing p. to p, and to <p (p -1 , we obtain the 

 formula, 



o=s P - 1 (^-p- 2 )-V- 1 , (b) 



as & symbolical form of the equation of Fresnel's wave. 



To interpret this equation, or to deduce from it a geometrical construction, we 

 may observe that the formula (assigned in the Seventh Lecture ), 



l = S(xpp, (c) 



is the equation of a certain auxiliary ellipsoid; and that 



<r=p- l Vp( P p = <Pp-p- 1 = (<p-p- 2 )p 



is a vector perpendicular to the plane of that diametral section whereof p is a semi- 

 axis. Hence 



= S<rp=So-(( P -p- 2 )- 1 <r 



is an equation which determines the two values of the square ( —p") of the length of 

 a semiaxis of the diametral section made by a plane perpendicular to a ; and if 

 T<r=Tp, so that the normal o- to the plane of the section is made equal in length to 

 one or other of the two semiaxes, then 



0=So-(<£-o-2)-V (d) 



But this is just the equation (b) of the wave, with <r written instead of p. Hence, 

 then, is at once derived the celebrated construction of Fresnel, namely, that " the 

 wave surface (for a biaxal crystal) is the locus of the extremities of normals to the 

 diametral sections of an ellipsoid, each normal having the length of one of the semi- 

 axes of that section." 



On certain Properties of the Powers of Numbers. 

 By J. Pope Hennessy, 31. P. 



On Gutta Percha as an Insulator at various Temperatures. 

 By Fleeming Jenkin. 



This paper contained an abstract of experiments, made for Messrs. R. S. Newall 

 and Co., to determine the absolute resistance of gutta percha, and the effect of tem- 

 perature on that resistance. 



The absolute resistance of gutta percha was calculated by the author from tests 

 on long submarine cables : the variation of resistance due to varying temperature 



