486 



Mr. Ft. Moon on the Integrahility of certain [May 16, 



111 assigning arbitrarily a function Y to the interior of S, in order to get 

 the internal density by the appHcation of the formula (1), we may if we 

 please discard the second of the conditions which Y had to satisfy at the 



surface, namely, that but m that case to the mass, of finite 



density, determined by (1) must be added an infinitely dense and infinitely 

 thin stratum extending over the surface, the finite superficial density of 

 this stratum being given by (3) . 



We have seen that the determination of the most general internal 

 arrangement requires the solution of the problem, To determine the poten- 

 tial for space external to S, supposed free from attracting matter, in terms 

 of the given potential at the surface ; and the determination of that parti- 

 cular arrangement in which the matter is VN^holly distributed over the sur- 

 face, requires further the solution of the same problem for space internal 

 to S. If, however, instead of having merely the potential given at the 

 surface S we had given a particular arrangement of matter within S, and 

 sought the most general rearrangement which should not alter the potential 

 at S, there would have been no preliminary problem to solve, since Y, and 

 therefore its differential coefficients, are known for space generally, and 

 therefore for the surface S, being expressed by triple integrals. 



Instead of having the attracting matter contained within a closed surface 

 S, and the attraction considered for space external to S, it might have been 

 the reverse, and the same methods would still have been applicable. The 

 problem in this form is more interesting with reference to electricity than 

 gravitation. 



11. " On the Integrahility of certain Partial Differential Equations 

 proposed by Mr. Airy.^^ By R. Moon, M.A., late Fellow of 

 Queen^s College, Cambridge. Communicated by Professor J. J. 

 Sylvester. Received April 30, 1867. 



(Abstract.) 



The equation 



0-^-a^^H-/3^+y.^ ..... (1) 

 dy~ dx' ay 



where «, /3j y are functions of x, includes two equations recently proposed 

 for solution by Mr. Airy, and affords a good illustration of the ordinary in- 

 capacity of partial differential equations of the second order for solutions 

 involving arbitrary functions. 



If the above equation admit of an integral solution containing one or 

 more arbitrary functions, it must be capable of being derived from an equa- 

 tion of the form 



F(.ry^)=?{/(^r)}. ........ ,(2) 



