66 Mr. Gwyther on a 



The solution is in this form. If (S.^?.^) are the components 

 of the displacement in a light wave from a point source 



where / stands for 27r/X, and u_x and ^^_i any two homo- 

 geneous functions whatever of degree- i, from which the 

 terms of other degrees are derived by the laws. 



?/_"2 + ^V^Z'-l = 0, w_3 + rv^^'-2 = 0, &C.J 



and the condition -^ + ^ + — = 0, will be satisfied provided 

 ax ay dz ^ 



x^v _i + y-yiV -I + s/u ^i = 0.J 



These expressions are to be used in this paper in their full 

 form without approximation, and I shall first show the 

 forms which the terms of other orders will take when 





where ?/„ is homogeneous of the 72"* order in x.y.z. 

 By continually using the formula 



2_ 



r"" y 



^V ^ra— ^m-\ " ^"*+l 



7^.2 = 





yU^I 



V *?/„ (« - I );^ V '^Un (^^ - I )n{ii-\- i)(/? + 2)?^^ 



+ 2 



.»+i ^"+3 



V ^/A, ( ;g-2)(;^-i)vX {n-2){n- \)n{n^ i)vX 



(^^j^2)^^^(^«jf3K 



■^ ^"+3 



The general term in the expansion being of the form 

 ^n-3+1 - ^ ^."-34-3 + ^'^^• 



+ ( + y 



./; -1-74-1 



