Longitudinal Component in LigJit. 265 



In every case the quartic cone intersects every plane per- 

 pendicular to the axis of oc in a bicircular quartic. 



In the case of a complex oscillator whose components are 

 not all parallel to the axis of #, as in the case just studied, 

 the longitudinal component will vanish to this order over a 

 quartic cone so long as we confint ourselves to a typical 

 term 8. This cone is of the form 



U=(* 8 +y a + *X<h<# + btf + c l z*)-\aa?{y + z)+bif(z + xj+c Z ^+y) 



+ ly*z < * + mz*x* + nxY 



+ xyz(px±qy + rz)). 



In general it is quite evident that the motion along the radius 

 does not vanish. 



On considering the general case, we may observe that if 

 the differentiations involved in 8 are such that for every term 

 « -1-/3 + 7 is either even or odd, then there will be a complex 

 surface all over which the normal component will vanish to 

 the second order of small quantities ; but that if a -f- /3 + 7 be 

 even in some terms and odd in others, we shall have cr of the 

 form 



d = 'U cos pt — qr + Y sm pt — qr, 



and this will only vanish over the curve of intersection of 

 U=0andV=0. 



IV. If we now consider the case of diffraction through a 

 narrow aperture, it is simpler to take the case of the electric 

 displacement of the incident wave as parallel to the edges. 

 In this case the electric force is everywhere parallel to the 

 edge, and consequently its longitudinal component every- 

 where vanishes. On the other hand the magnetic force is 

 perpendicular to the slit and has a longitudinal component 

 everywhere except in the plane through the slit perpendicular 

 to the wave-face. In considering the more complicated case 

 of the electric displacement being perpendicular to the slit, it 

 is necessary to take account of the nature of the edges, whether 

 they are non-conductors or conductors, whether they are 

 crystalline, and so forth, because their electrification <fec. 

 must come into consideration. Similarly, in the case of the 

 electric displacement being parallel to the slit the magnetic 

 properties of the edges may he important. In this case, too, 

 their conductivity influences the effective width of the slit, as 

 is evidently the case when we are dealing with wire gratings 

 in the path of the Hertzian radiations. These questions are 

 involved in a complicated way in the whole discussion of the 

 effect of a grating on the plane of polarization of the incident 



