50 



Mr. Mac Cullagh on the Laws of 



V. 



Draw the great circle L,K, at right angles to T,P,, and meeting it in K, ; 

 then the plane of L,K, will be the plane of the transversals, since the latter plane 

 passes through L^, and is perpendicular to T,P,. But the tangent of P,K, is 

 equal to the tangent of P,L, multiplied by the cosine of the angle P, or by the 

 sine of 6^; therefore, denoting P,K^ by e,, and recollecting that P,L,=k, we find, 

 by (28), 



(29) 



lane, 



sin'^t^ 



tan£ 



-sin% 



Now we have seen that the ratio of OP to OS, or of OS to OG, is the index of 

 refraction; so that sin^, is to sin^t^ as OP to OG. Therefore, by (29), 



tanE,_ OG _OG 

 GP' 



(30) 



tane OP-OG 



but OG is to GP as the tangent of the angle GPT is to the tangent of the 

 angle GOT j and since e is the angle GOT, it follows that e, is equal to the 

 angle GPT or KOP. Consequently, OK will meet the surface of the sphere 

 in the point K,. Thus we have proved our assertion, that, when there is only 

 one refracted ray, the plane of the transversals is the polar plane of that ray. 



The sign of the quantity h is always the same as that of the cosine of the 

 spherical angle T,P,Y,. But to remove all ambiguity respecting signs, we must 

 make a few additional conventions. Supposing, as we have hitherto done, that the 

 refracted light moves from O to T, and conceiving a right line to be drawn from 

 the origin parallel to GT, and directed from G towards T, let the angle S^, 

 which this right line makes with the plane of incidence, be reckoned, like 0,, 6^, 

 from an initial position comprised between the negative directions of x and y ; 

 and let ^^, like the angles 6^, 0^, 0^, increase on the side of z positive, and range 

 from 0° to 360°. Then S^ will always be equal either to the angle P, of the 

 spherical triangle T,P,Y,, or to the reentrant angle, which is the difference 

 between P, and 360". In either case, the cosine of 3-^ will be the same, both in 

 magnitude and ^gn, as the cosine of the angle T,P,Y,. Consequently, if, 

 instead of (25), we use the direct trigonometrical formula 



cos t™ =cos tgCos £ -j- sin i^ sin t cos ^^ 



we shall find 



(8) 



(31) 



siiV^igtan e cosS^j 



sin^tfg 



X 



/ 





