FORM OF THE WAVE SURFACE. 



225 



pupposition, Mr. Fresnel obtained an equation for the wave surface, wlilcli is the 

 followinc; : 





2_.(«2 + ,2)] + 



\a^+h^)\^d'h\^=^. [50. 



Or, (^ 



„2. 



This is an equation of 



the fourth degree, and 



^^ represents a surface of 



■^ ' .iMIIiir'l^s. ^ two nappes, or sheets, 



inosculating at four 

 points. Figure 64 is a 

 representation of this 

 surface copied from a 

 drawing made by Mr. 

 Ferdinand Engel, of 

 Washington city. In 

 order to exhibit the in- 

 terior nappe, two ungu- 

 l.ie are represented as 

 cut away; one of the 

 section planes passing 

 through the two points 

 of inosculation in the 

 visible surface, and an- 

 other through one of 

 them. 



The form of the wave 

 being known, we may 

 apply, for the detenni- 

 nation of the direction 

 ^^' * of a ray, the principle 



on which Huyghens founded his construction for spherical and spheroidal waves. 

 Resuming once more the figure employed in illustrating that construction, we 

 may say let CD be the direction of the semi-axis of 

 elasticity a — the semi-axes h and c being at right 

 angles to this, and to each othei*. Upon these axes 

 let the wave surface be constructed in space, with C, 

 the point of incidence, as the centre; the values of a, 

 h, and c being the velocities of rays moving at 

 righr angles to them (and whose molecular movements 

 Yi-y. 34. are therefore parallel to them) when the velocity in 



vacuo is made unity. If MN be the surface of re- 

 fraction, RC incident ray, and CP the normal to the surface, then RCF is the 

 plane of incidence. In this plane draw CG perpendicular to RC. CG will be in 

 the incident wave front. ]\Iakc RC=cl, draw RG parallel to the refracting sur- 

 face, and cutting CG in G. Draw also GQ parallel to RC. Then when the 

 Avav(! front has advanced to Q it will intersect MN in a line drawn through Q 

 perpendicular to the plane of incidence. If the plane of the diagram be sup- 

 posed to be the plane of incidence, tlats perpendicular will be projected mto the 

 point Q. Both the refracted waves will intersect MN in the same line; and their 

 planes will be also tangent to the two sheets of the surface. If ADB represent 

 one of these sheets, and HFK the other, then tangent planes passing through 

 the perpendicular projected in Q and meeting these sheets, as at E and F, will 

 determine the directions, CE, CF, of the refracted rays. It is to be observed, 

 however, that the points E and F, and therefore the refracted rays CE, CF, 

 15 s 



