66 Mr. Mac Cullagh on the Laws of 



reflexion at the second surface of a crystal ; but these must be reserved for a 

 future communication. It would be easy, Indeed, to write down the algebraical 



refracted ray K;,, emerging from the crystal in a direction parallel to b,. Put r,, r^, Tj, and 

 f ^tJ , r/,) for the transversals of the rays in the order in which they have been named. As the 

 transversal r is supposed to be given in magnitude, the lengths as well as the directions of r, and r, 

 can be found by the construction in page 40. 



Now, the direction of r^ being changed, and its magnitude retained, let the ray R, be turned 

 , directly back, so as to be incident again on the crystal, and to suffer reflexion and refraction at the 



first surface. Then the two refracted rays which it gives will be parallel to B,,, b',,, and their trans- 

 versals will be equal and parallel to r„, t'„. The reflected ray which it gives will coincide with 

 R, ; and the reflected transversal, when compounded with rj, will furnish a resultant equal and 

 parallel to the emergent transversal r(,). 



Thus the Tsonstructions, which have been given for the first surface, may be made available for 

 the second surface, and every question relative to crystalline reflexion may be solved geometrically 

 by means of the polar planes. 



The foregoing rule was not, properly speaking, deduced from theory. I first formed a clear con- 

 ception of what the rule ought to be, and then verified it for the simple case of singly refracting 

 media, and finally proved it for doubly refracting crystals. The truth of the rule, in crystals, 

 .depends upon the truth of the three following equations : 



sin(j„+i'„) |cos(i,— i'„)4-cotane„cotan3'„| ^h„ + h'„ = 0, 



siH(ij — (,,) |cos(i, -|-(,,) — cotanfl, cotanS,, j -\- h, — A,,:=0, • (viii.) 



sin((s — (',,) Jcos(;i-4-i',,) — cotanSjCotanC',, | 4- /», — A'„=:0, 



in which the notation is intelligible without any explanation. The first equation is the same as 

 equation (vii.) already noticed; and the other two differ from it only in appearance, the change in 

 the signs being occasioned by a change in the relative position of the rays. 



When the reflexion is total, I suppose we may follow the example which Fresnel has set us in the 



; case of ordinary media. The general algebraic expression for each reflected transversal will then 



I become imaginary ; and by putting it under the form 



I T (cos <J) -f- -y/ — Isin*), 



j we shall have t for the reflected transversal, and * for the change of phase. 



; From the nature of the rules which we have given for treating the question of reflexion at either 

 surface of the crystal, it follows that the final equation, for determining the position of a transversal, 

 is always linear, though the equation of vis viva is of the second degree. This result verv strongly 

 confirms the theory; but it shows, at the same time, t|iat the law of the preservation oivisvivtt, is 

 not to be regarded as an ultimate principle, but rather as a consequence of some elementary law not 

 yet discovered. 



It now appears that the conjectures put forward in the note, p. 36, were hasty, and that there 

 was some mistake in the calculations which gave rise to them. It is scarcely necessary to mention. 



