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XXIII. OK TOTAL BEFLEXION. 



Proceedings of the Royal Irish Academy, VOL. in. p. 49. Read, January 13, 1845. 



PROFESSOR MAC CULLAGH made a communication on the subject 

 of Total Eeflexion. 



In the case of total reflexion the vibrations which take place 

 in the rarer medium are in general elliptical, and when this 

 medium is a crystal, the equations by which the ellipse of vi- 

 bration is determined are very complicated. The projection 

 of this ellipse upon the plane of incidence may, however, be 

 easily found by the remark in p. 12 of the present volume ; 

 the projecting cylinder is therefore known, and as the ellipse of 

 vibration is a section of this cylinder, the question of deter- 

 mining the ellipse is reduced to that of determining its plane. 

 For this purpose Mr. Mac Cullagh gave the following rule. 

 Having constructed the ellipsoid of indices (that whose axes 

 are parallel to the axes of elasticity, and inversely proportional 

 to the three principal velocities of propagation in the crystal), 

 let its two planes of circular section intersect the aforesaid 

 cylinder. The curves of intersection will be ellipses, which 

 shall be supposed to have a common centre in the axis of 

 the cylinder. Let OP, OP' be the greater semiaxes of these 

 ellipses, and OQ, OQf the less semiaxes; the lengths of the 

 two former being denoted by p, p', and the lengths of the two 

 latter by q, q'. Join the extremities P, P' of the greater 

 semiaxes, and the extremities Q, Q! of the less semiaxes; and 



