150 



Sir William Thomson on the 



passing through the point (x,y, z), and by ^, ^, J-, — dif- 

 ferentiations respectively on the supposition of x, y, z constant, 

 t, y, z constant, t, x, z constant, and t, x, y constant, the ordi- 

 nary equations of motion are 



dp dx , . dx , . dx , . dx 1 

 dx dt dx * dy 



dz' 



dy-dt +0C Tx +y ~di, Z dz> 



dy dt 

 dp _ dz 



and 



. dz . dz dz 

 dz dt dx dy dz' 



dx dy dz _ 

 dx dy dz 



To transform to the columnar coordinates, we have 

 x = r cos 6, y = r sin 0, 



x = r cos 6 — rd sin 6, 



y = r sin 6 + rd cos 0, 



d a d • a d 



— - = cos -j sm V —Trf 



ax dr rau 



d . d d . a d \ 



-i-= sm6' -r- + cost 1 — -^. J 

 dy dr rdo 



The transformed equations are 



dp dr , . dr (rd) 2 , a c/r , . dr 

 -ah=dt +r ^- "+'-** + 



( /r 



rf<9 



rfs' 



rdO T dt^' dr ^' ^ dO~ +Z ~dz~> 

 __ dp _d,z dz Adz dz 

 dz"~dt +r dr~ + d0+ Z dz> 



and 





a) 



(2) 



(3) 



W 



(5) 



Now let the motion be approximately in circles round Oz, 

 with velocity everywhere approximately equal to T, a func- 

 tion of r ; and to fulfil these conditions, assume 



r = p cos mz sin (nt — id), rd = T + r cos mz cos (nt — id), 

 z = iv sin mz sin ( w£ — iff) , p = P + otcos w?2 cos (nt — id), [ . 

 with ps= f T 2 ^ 



