172 VORTEX MOTION. [CHAP. VI. 



of the one in advance, and to contract the radius of the one in 

 the rear. If the radius of the one in front become larger than 

 that of the one in the rear, the motion of the former ring will 

 be retarded, whilst that of the latter is accelerated. Hence if 

 the conditions as to the relative size and strength of the two 

 rings be favourable, it may happen that the second ring will 

 overtake and pass through the first. The parts played by the 

 two rings will then be reversed ; the one which is now in the 

 rear will in turn overtake and pass through the other, and so on, 

 the rings alternately passing one through the other. 



If the rotations in the two rings be opposed, and such that 

 the rings approach one another, the mutual influence will be to 

 enlarge the radius of each ring. 



If the two rings be moreover equal in size and: strength, the 

 velocity of approach will continually diminish. In this case the 

 motion at all points of the plane which is parallel to the two 

 rings, and half-way between- them, is tangential to this plane. 

 We may therefore, if we please, regard this plane as a fixed 

 boundary to the fluid on either side of it, and so obtain the 

 solution of the case where a- single vortex-ring moves directly 

 towards a fixed rigid wall. 



On the Conditions for Steady Motion. 

 145. In steady motion, i.e. when 



d u _ A av _ r\ dw _ ~ 



the equations (2) of Art. 6 may be written 



du dv dw %(*&amp;gt;__ \_ &V 1 dp o 



Hence if we make 



P=|^E + F+i 2 (52), 



we have 



=2( W f-ur), 3; 



