1881.] Mr GreenMU, On height consistent with stability. 67 



To solve this differential equation, first put p = oo*z, then 

 2 d 2 z , dz ( w 3 , \ 



Again, put # 3 = r a , and then 



2 d 2 ? ffe _ g / 2 d 2 £ <fe\ 



and therefore 



„ d 2 z dz ( 4>w „ 



This is of the form of Bessel's differential equation 



r2 £ +r ^ +(KV - ?i2) " =o (3) ' 



where k 2 = ^ 2 , n 2 = ^ . 



The solution of (3) is 



z = JJ n (Kr) + BJ_ n (,cr), 



x n f n 



where J n (x)= cos (x cos</>) sin 2n <^(£. 



V7TZ J. (tt + 2VJ0 



(Todhunter, Functions of Laplace, Lame, and Bessel, p. 285.) 

 Consequently the solution of (2) is 



z = AJ^(icr) + BJJ/cr), 

 and the solution of (1) is 



p = »Jx {AJfox*) + BJ^kx*-)). 



The condition that ~ = when x = 0, makes .4=0, and then 

 ax 



p = B *JxJ_ ^(/cx^) (4) . 



At A, the lowest point, we must have p = 0; and therefore, 

 supposing the height OA to be k, 



If c be the least root of the equation J Ac) = 0, 

 then c = kJi-, 



'-er-pso*. 



5—2 



