282 Mr R F. Gwyther, On the solution of the [May 25, 



Hence i 

 z of degree 

 by the law 



Hence u_ x and v_ t may be any homogeneous functions of x, y and 

 z of degree — 1, and the terms of other degrees are to be derived 



By this means we may obtain a complete formal solution 

 suitable for values of r greater than \, for on examination the 

 orders of magnitude of the consecutive terms are to be compared 



with those of the geometric series 1 . - . -§ , etc. Hence the series 



in the case of light may be considered to be limited to the first 

 term, and would actually terminate provided vV_„ = 0. 



In the general case we notice that if u_ x and v_ t are divided 

 into any number of parts to suit any linear algebraic or linear 

 differential condition ; and if each of these parts is expanded by 

 the law above, the sum of the separate expansions will be the 

 original expansion. 



Thus if we write £ = £ + k ( -~ — -j!-\ , etc., we should find that 



for any value of k this division can be always made, and in one way 

 only. 



If, moreover, we want two similar expansions for rj and £", where 

 £, 7), £ are to satisfy the relation 



dx dy dz 

 we may obtain them without loss of generality, thus 



* dy dz ' dz .dx ' dx dy ' 



where £ , 7] , £ are unconditional solutions 



f = % {A t sinp (r — at) + B t cosp (r — at)}, 

 7] = S {A 2 sin p (r - at) + B 2 cos p(r- at)}, 

 £ = 2 {A 3 sin p (r — at) + B 3 cos p (r — at)}, 



and f = 2 {C^ sin p (r — a£) + i^ cos^j (r — ai)}. 



I have shewn how to deduce A and B from the first terms (say 

 a and b). I will shew how to obtain the corresponding first terms 

 in G and D (say c and fT). The following terms will be found by 

 the same rule as before. 



