the Luminiferoiis jailer. 345 



from the common equations, weie stable, the motion which 

 would be obtained from equations (1.) would approach inde- 

 finitely, as jw. vanished, to one for which ndx-{- ... was an exact 

 differeniial, and therefore, for anything proved to the contrary, 

 the latter motion might be stable; but if, on the contrary, the 

 motion obtained from (1.) should turn out totally different 

 from one for which udx-\- ... is an exact differential, the latter 

 kind of motion must necessarily be unstable. 



Conce've a velocity equal and opposite to that of the sphere 

 impressed both on the sphere and on the fluid. It is easy to 

 prove that udx+... will or will not be an exact differential 

 after the velocity is impressed, according as it was or was not 

 such before. The spheie is thus leduced to rest, and the 

 problem becomes one of steady motion. The solution which 

 I am about to give is extracted from some researches in which 

 I am engaged, but which are not at present published. It 

 would occupy far too much room in this Magazine to enter 

 into the mode of obtaining the solution : but this is not neces- 

 sary; lor it will probably be allowed that there is but one 

 solution of the equations in the case proposed, as indeed rea- 

 dily follows from physical considerations, so that it will be 

 sufficient to give the result, which may be verified by differ- 

 entiation. 



Let the centre of the sphere be taken for origin ; let the 

 direction of the real motion of the sphere make with the axes 

 angles whose cosines are I, m, n, and let v be the real velocity 

 of the sphere ; so that when the problem is reduced to one of 

 steady motion, the fluid at a distance from the sphere is moving 

 in the opposite direction with a velocity v. Let a be the 

 sphere's radius : then we have to satisfy the general equations 

 (1.) and (2.) with the particular conditions 



« = 0, v = Q, 'K) = 0, when ?• = « ; (3.) 



71=.— Iv, v= —mv, to =—«)/, when r= oo , . (4.) 



r being the distance of the point considered from the centre 

 of the sphei'e. It will be found that all the equations are 

 satisfied by the following values, 



3 a 



7^= n+ 2 /*" ^3- (^-^ + w'i/ + «-)j 



with symmetrical expressions for v and w. U is here an arbi- 

 trary constant, which evidently expresses the value of j) at an 

 infinite distance. Now the motion defined by the above ex- 



