426 MOTION. 



velocity of the centre of the wing would be the descent in the wing were constant, the number 

 same in all birds, and if the arc of vibration of flappings would vary inversely as the dis- 

 arid 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,* t = the time reckoned from the mo- 

 ment when the wings begin to be depressed, r = the time of a depression, T' = the time of an 

 elevation, W the weight of the bird, if = the weight of a cubical foot of air, i? r= the velocity 

 acquired by gravity in an unit of time, k = a constant coefficient, by which we multiply the product 



s - n, in order to get the expression for the resistance of the air to the rising of the bird, which 



**O 



is therefore it k s 5 K = a similar coefficient for the expression of the resistance of the air to the 



2# 



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 



M 2 



weight, and by the resistance of the air to its rising, namely, it k s ^ > but it is driven upwards by 



'""'O 



the resistance which the air opposes to the motion of the wing, that is, it K A \ " hence 



2^ 

 the equation to the motion will be 



g dt 2ff 2g 



or 



2 W =. it K A (Vu-f7r ksu 2 2 Wg- ....................... (12) 



' ' t 



These are approximative values. The motion of the wing being very quick, we may consider u to 

 be constant during a depression, and if u , w, be the values of u, at the beginning and end of the 

 depression, equation (12) will become 



2 W (,-)= T{arK A (V-if ) 2 ir fcstig 2 W g-} ............... (13) 



Similarly if H t , w 2 be the values of u at the beginning and end of an elevation, the equation for the 

 motion will be 



2 W (, ,) = "'{* K'A' (V + M,) 2 + fc s MI =+ 2 Wg-} ............... (14) 



By adding the two latter equations together we have 



2 W( a )=T{*rKA(V ) 2 TT k s wg-2 Wg-} ^{w K'A'( V'+w^'+w fc s u 2 +2 W g-} ....... (15) 



Since, at the end of every flapping the wings are supposed to be in the same position as they were 

 at the beginning of it, we must have 



V T = V T' ....................................... (16) 



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 2 u which the velocity M undergoes in the 

 time T -\- T'. If we suppose the motion of the bird to be uniform during each flapping = j = i/ 2 , 

 and if we take the value of V in equation (16), equation (15) will become 



= TT' *r K A (V 1/ ) 2 wK'A'(Vr + K T')2 (T+T') T' k s u<?(r + T') T' 2 W g-. ..(17) 



In order to fly, the bird must expend a quantity of force sufficient to overcome the resistance of the 

 air to the motion of the wings, and, considering the velocity of the bird to be constant during each 

 movement of the wings, the quantity of force expended in a depression will be 



and in an elevation 



If we now suppose M O = w, as before, and substitute the value of V, we shall have for the whole 

 force expended in an unit of time 



(18) 



Let T' = )> T, and K'A' = q K A, equation (17) then becomes 



If S 2 W 



o = P ( v-tO'-g (V + P? - ( P + P 2 ) o 2 - (p + ;> 2 ) 



* V and V arc supposed constant during the times of elevation and depression, respectively, and are referred 

 to the centres of the wings. 



