544 Lord Rayleigh on Convection Currents in 



and for the boundary condition, setting ka — z and omitting 

 the argument in the BesseFs functions, 



a { j \ cos <£ 4- kp j " + pyjy"} 



+ { J n ' + kp J n "\{An cos nO + B n sin n6} 



~ J n ( ~~ ^-» s i n n @ + B n cos 7l ^} { """"<*» s i n ^ 



<z£ 



+ &COS710}=:O. . (61) 



If for the moment we omit the terms of the second order, 

 we have 



A J(/ + kA J ' f {a n cos n6 + fi n si n n @\ 



+ J w '{A»cosn0 + B Jl sinn0}:=O; . (62) 

 so that 3 f (z)—0, and 



M. J ''. a „-f J n '.A^0, MoV.A + J/.B^O. . (63) 



To this order of approximation z, = ka, has the same value 

 as when p = 0; that is to say, the equivalent radius is equal 

 to the mean radius, or (as we may also express it) k may be 

 regarded as dependent upon the area only. Equations (63) 

 determine A*,, B TC in terms of the known quantities a n , fi n . 



Since J ' is a small quantity, cos (j> in (61) may now be 

 omitted. To obtain a corrected evaluation of z, it suffices 

 to take the mean of (61) for all values of 0. Thus 



A {2J ^ + P 2 Jo" / K 2 + /3, 2 )} 



+ { kJ n " - n 2 J n /az}{* n A n + fi n B n } = 0, 



or on substitution of the approximate values of A n , B n 

 from (63), 



This expression may, however, be much simplified. In 

 virtue of the general equation for J n , 



n 2 J ' 



and since here , " , J ( / = approximately, 



Thus 



J '(*) = i* s J .2 («„ 2 + A, 2 ) {ji + ~}, • (65) 



the sign of summation with respect to n being introduced. 



