426 MOTION. 
velocity of the centre of the wing would be the descent in the wing were constant, the nur ber 
same in all birds, and if the are of vibration of flappings would vary inversely as the dis- 
and the ratio of the time of ascent to that of tance of the centre of the wing from the axis of 
wings, V‘ = the velocity with which it elevates its wings,* ¢ = the time reckoned from the mo- 
ment when the wings begin to be depressed, + = the time of a depression, +‘ = the time of an 
elevation, W = the weight of the bird, r = the weight of a cubical foot of air, g = the velocity 
acquired by gravity in an unit of time, & = a constant coefficient, by which we multiply the product 
2 
=" in order to get the expression for the resistance of the air to the rising of the bird, 
is therefore wk s = K = asimilar coefficient for the expression of the resistance of the air to th 
& 
depression of the wings, K’ = a similar coefficient for the resistance of the air to its elevation. 
During the depression of the wing the bird is drawn downwards by the force W, arising from its own 
weight, and by the resistance of the air to its rising, namely, w k s but it is driven upwards | ; 
the resistance which the air opposes to the motion of the wing, that is, r nate, hence 
the equation to the motion will be 
Wok AsO? ets em 
g dt 2g 2g 
7 8 
rf 
2W ark A (V—u)?—m k 8 u2—2 W g..erccccccccessecsseees(la) ‘ 
These are approximative values. The motion of the wing being very quick, we may considers to 
be constant during a depression, and if u,, u, be the values of u, at the beginning and end of the 
depression, equation (12) will become Be 
4 WwW (u,—u,) = v{ T KA (V—tu,) 8 ane T ks 13 —2 W g}. os cokes lee ) 
Similarly if w,, u, be the values of u at the beginning and end of an elevation, the equation for he 
motion will be 7 
“}) 
2W(u,—u,) =—7{r KA’ CW + u,)*+wrksuZt+2 Wg}. ee eveeeere rd © is 
By adding the two latter equations together we have 
QW(uy—u,)==t{ mK A(V—uy)*—a ke 8 ug —2We}—r' {we K'A( V0, 2-9 ks u2+-2 W gece eens a5) 
Since, at the end of every flapping the wings are supposed to be in the same position as they wel 2 
at the beginning of it, we must have ; a 
Vr=V' Veveaccccccerssccccccersccneresesesecssel 
The two equations (15) and (16) express the conditions under which the bird may acquire a given 
motion, the nature of which is shown by the variation u, — u, which the velocity u undergoes in the 
time + + +. If we suppose the motion of the bird to be uniform during each flapping u, =u, = nt 
and if we take the value of V’ in equation (16), equation (15) will become : 
=tr er KA (V—u,)—aK' A’ (Ve+ ur’ P4417) 7 wk s u2—(7r +7) r 2Weg. .. 7) 
In order to fly, the bird must expend a quantity of force sufficient to overcome the resistance of hi 
air to the motion of the wings, and, considering the velocity of the bird to be constant during ea 
movement of the wings, the quantity of force expended in a depression will be 
V—un,)2 7 
) wK ace Ve, 
and in an elevation : 
wk AW +4) ye, 
2g 
If we now sup u, =u, as before, and substitute the value of V’, we shall have for the wh 
force expended in an unit of time oS 
erat KA(V—u,) + Ka (tery : . odbt ge 
, 
(1 
pie 
Let +’ = p+, and K’A’ = q KA, equation (17) then becomes r 
“Ts 
0 =P Vu) —4 (V +p —@ +P) ES g—@+e) ANH 
aKA —" 
* V and V’ are supposed constant during the times of clevation and depression, respecti and are referr 
to the centres of the wings. ; be f Sir 7 a 
"Ta 
