MaiicJiester Memoirs, Vol. xliv. ( 1 899), No. ;>. 5 



everywhere horizontally but with a v^clocity increasing 

 upwards, taking for the sake of simplicity the imagi- 

 nary case of a wind uniform above and below a certain 

 plane where the velocity changes. Since a uniform motion 

 has no effect, we may suppose without further loss of 

 generality, that the velocities of the wind above and below 

 the plane are -\-u and — //. Let us consider how a bird, 

 sailing somewhat above the plane of separation and 

 endowed with an initial relative velocity v, might take 

 advantage of the position in which he finds himself 



The first step is, if necessary, to turn round until the 

 relative motion is down wind (in the upper stratum) and 

 then to drop through the plane of separation. In falling 

 down to the level of the plane there is a gain of relative 

 velocity, but this of no significance for the present purpose, 

 as it is purchased by the loss of elevation ; but in passing 

 through the plane there is a really effective gain. In 

 entering the lower stratum the actual velocity is indeed 

 unaltered, but the velocity relatively to the surrounding 

 air (moving in the opposite direction) is increased. 



\i h denote the height above the plane of separation 

 to which the initial relative velocity v is due, we have 

 i>' = 2gh. Here i' is the velocity, relatively to the air in 

 the upper stratum, with which the bird crosses the plane. 

 After crossing, the velocity, now reckoned relatively to the 

 air in the lower stratum, becomes v-\-2u, and the new 

 value of h is given by 



2g/i = (?'+ 2l(Y, 



so that 



Here iji -Ji) is the gain of potential elevation and, if u is 

 given, it increases as v increases. 



At this stage the bird is moving against the direction 

 of the wind in the lower stratum. He next turns round — 



