on the Resistance of the Atmosphere to an oscillating Sphere. 323 



du \ . 



"^ " dlc'^ ) ^^ ^ quantity of the second order: the equa- 

 tions are reduced to these : 



And the method of testing the possibility of any assumed 

 system of movement, or of discovering the conditions which 

 must be satisfied by the system of movements, will be, to form 

 the expressions on the right-hand side of these equations, and 

 to try whether they satisfy the equations which must hold 

 among the four differential coefficients of the same quantity. 



I. Now suppose the whole velocity of the particle x, y, z, 

 to be directed from the centre, and at the time t to have the 

 value V (motion towards the centre being implied by a nega- 

 tive value of u) ; and suppose that centre to be the origin of 

 coordinates. Then 



X y Z , 



ti - 'J--, V = tj~-, -w =: V — : where r = ^ a;«+y + ^s. 



du _ x_ dv_ dv _ y du dvo _ z dv 



dt ~ r' dt'TT - ^' ~dt' IT ^ T'Tt' ^'^° 



du^ __dv_ ^ /J x_ dr\_du^ x /I :^\ 



dx dx ' r \r r^ ' dx )~ Tx'T^ "^ \r'~'?)'' 



_^_ _^ y_,^(}_ 2t\. d^_dv z n ^2x 



dy dy'r^'^Kr r^J ' dz~ dz'T'^" V~r~ TV* 



du dv dw I f du dv dv\ 2 u ^ , 



rfiT + 5;^+ Z7 = rV^z:; + 3/^ + ^ rf^) + — • Sub- 



stituting these values in the equations above, they become 

 d . log p __ kx dv 



dx ~ r~ ' ~dJ ~ 



d.logp _ _ ky_ d_u_ _ 



dy r ' dt "" ^ 



