and the Dynamical Theory of Diffraction. 415 



Spheeical Waves. 



Let F , Gr > Ho 5 I\, G x , &c, be a system of vectors derived 

 from one another by the equations 



FCt±± m Cl\j m 

 m+1 — " 



dy dz 



p _ dF m __ dH m 

 ^+ 1 ~ <fc dx 



jj _ dG m d~F m 



dx dy 



Then, as is well known, all this system of vectors necessarily 

 satisfy the equation of continuity, 



d¥ m dQ m + dK m 



dx dy ' dz ' 



except the primitive ones F , G , H , and even these can be 

 made to satisfy the equation by the addition of terms of the 

 „ cl3 dJ dJ 

 dx' dy' dz' 

 The equations of light- waves, either on the electro-magnetic 

 theory or the elastic-solid theory, are of the form 



dr I dx ) dx 1 



^ — V 2 -f A 2 f2- — — "I + « 2 — 



dt 2 I dz J dz' 



J= cU^ clG dR 



dx dy dz * 



dx 2 ^ dy 2 ^ dz 2 ' 



In the electro-magnetic theory v is always zero. In the 

 elastic-solid theory the terms in v give the wave of normal 

 disturbance like that of sound, and can nearly always be 

 omitted, as the transverse and normal waves are immediately 

 separated when v is different from V, as it always must be* 

 As we wish to use the derived functions only, the terms in J 

 can also be omitted, as Maxwell has shown ; and hence we 

 can write, on either theory, 



2F 2 



