Prof. Challis on the Force of Gravity. 445 



sensible error they may be omitted. Also it appeared from the 

 reasoning that the propagation into space of the condensations 

 due to the disturbance of the original waves by the reaction of 

 the surface of the atom, might be regarded as instantaneous. 

 Hence, P being put for Nap. log p, and its values being restricted 

 to those condensations, it follows that 



'P = m<}>{x, y, z) sin^-^— +cj, 



which form of expression is plainly consistent with the hydrody- 

 namical equation 



dt^-" '\dx^ "•" dy^ '^ dz" )' 



Hence the supposition that V=f[t)^{x, y, z), which was made 

 in the former communication, is shown to be legitimate, the 

 function /(/) satisfying the equation f"{t) + «y(0 = 0- But the 

 above equation, being linear, allows of making a supposition 

 more general and more applicable to the proposed problem, viz. 



P=/i(0<^i (*". y> ^) +fS)<f>^ {^, y, -) + &c., 



provided that f>'{t) + u%{t) = 0, //(O + cc%{t) = 0, &c. For if 

 these equations be satisfied, it will be found that 



d^ d^ d^ «^P_ 

 dx^^ dy^'^ dz^ '^ b^ ' 



0? 



or, putting Ti^ for j-^, and transforming to polar coordinates, 



-^+-.(-^g^ + ^^cot^)+AVP = 0. 



To obtain a particular solution of this equation, let it be assumed 

 that 



d.r? 



-^ = (/i^j +/2'f 2 cos 0) sin e, 



A ^^^A being, by what is shown above, periodic functions of 

 the time, and ■^|r^, -yjr^ being by supposition functions of r only. 

 Then substituting in the above equation, after differentiating it 

 with respect to 6, there will result 



Consequently the equation is satisfied if -i/r, and ^}r^ arc the par- 

 ticular functions of r given by the integration of the equations 



