10 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



It is to be observed that •&', tst", nr" are not independent. For differentiating equations (13) 



with respect to x, y, z, and adding, we get 



d-ar d-ar" dw"' , v 



dm dy dz ■ 



Hence •&', ■&"> w" must be supposed given arbitrarily only in so far as is consistent with i 

 the above equation. 



Eliminating £from 0*)> ano ^ ^ e second of equations (13), we get 



d fdf dr,\ d?£ dsr" 

 1 + — t = 2- 



d td% dr)\ 

 Ix \dx dy) 



dx \dx dy) dz 2 dz 



„ d Idn df\ dip" 



or v I +-T- h -- -r\ = 2 ^r~' 



dy \dx dy) dz 

 which becomes by the last of equations (13) 



Consequently, by equation (5), 



Idsr" dw"\ d V 



i-hm 



dy dz j v 



Transforming this equation in the same manner as (11), supposing x, y, z measured from O, 

 and writing down the two equations found by symmetry, we have finally, 



£=^///(^"'-*^")^ 





(16). 



. 7. Let d, tst', w", "&"' be as before ; and. let it be required to determine three functions 

 £, rj, £ from the equations (9) and (13). 



From the linearity of the equations it is evident that we have merely to add together the 

 expressions obtained in the last two articles. 



8. Let £ , r) , £ be three functions given arbitrarily within a finite space outside of which 

 they are equal to zero: it is required to decompose these functions into two parts £„ jj„ £, and 

 £21 »?a. £2 suc h tnat %\dx + r) x dy + £i4m may be an exact differential d\|/„ and f 2 , t) 2 , £ 2 may 

 satisfy (14). 



Observing that £ 2 = £, - £, ~ = % - ^ £ 2 = £ - £„ expressing (•„ m , £, in terms of f u 

 and substituting in (14), we get 



where S is what 5 becomes when £ , t; , £ are written for £, >;, £. The above equation gives 

 whence £„ >;„ £„ and consequently £ 2 , % , £ 2 , are known. 



