64 ProceedingH of tJic lloyal Iris// Acadrmf/. 



a result which may be written in the following form. Let 

 be the original kinetic energy, = a,ji the roots of Ji {x) = ; 

 then 



This verifies, for ^ = 0. For we know that 



Decay of Areal Motion. 



Let A be the total moment of momentum at any time. We 

 find in the same way 



III. Vortex spreading in Spa§e. 



Given initially, = from r = to r = a, and wr^ = fiflt' from 

 r = a to r = R, while for all values of if, w = for r - R. 

 Consider, first, the expression for ^. Write 



^ 2/ 



^ = h-V%c,r,e P '/o(a„/). 



'Now, any function ^ of may be expressed from r = to r = Jt in a 

 series in J^^{a,,^r), the a,„ being given by Ji{a„^R) = 0. 



Now, putting ^ = 0, we have ^ = X' + ^c^JJ^a^nr) ; and this 

 O 



is to represent — from r = to r = a, and from r =^ a to 

 r = R. We have then 



R 



- J,Xa,,R) = J r(J,{a^r)dr = -J rJ^ia^r) dr, 



