428 Prof. M. F. FitzGerald. 



Dropper " seem to show that this is by no means the case, but, un- 

 fortunately, he was unable, through uncontrollable circumstances, to 

 determine the proper value, which is some function of the velocity, and 

 may possibly be so small as g/3 at his highest velocities. If so mV 

 which is the minimum value of W requires corresponding reduction, 

 and, in the case taken, the scale of W is about three times too large, 

 reducing the necessary work to about 1 ft.-lb. per pound carried or 

 thereabouts. It is now evident how the energy expenditure of birds 

 may lie within quite reasonable limits, though we cannot, with any 

 approach to accuracy, make the calculations for the design of a flying 

 machine with flapping wings. We can say it will not in a given case 

 cost a pound an hour for power, but we cannot tell whether it will 

 cost only sixpence, or as much as half-a-crown or three shillings. 



To allow for head resistance of the bird's body some term must be 

 added to the left-hand side of equation (A), which will alter the 

 absolute term of equation (C) and the J undej the square root in equa- 

 tion (E) so as to increase W, and a number of small corrections of 

 other kinds might be made, which, however, considering the roughness 

 of the numerical data we have to go upon, are not worth taking into 

 account. 



The justification of the assumptions as to the smallness of a and s 

 is best exhibited by a numerical example. Taking then AV =120 

 with A = 2 square feet per pound of bird, and assuming S = 0'5 (a 

 stroke of 1 foot) we find that if p = 30 (about 300 downward strokes 

 per minute, which sea-gulls in regular flight often exceed) and cos = 

 1, Spcos0 = 15, and W = 2'65; while /* is 0'35, and m is 0-041. 

 The maximum value of a is then 0'041(1 + 0'35) when pt = IT. 

 This is 0-0554, the circular measure of 3 10' nearly. The maximum 

 value of s, the slope of wing path, is ^?S/V = 0'25 ; the circular 

 measure of 14 20'; but these do not occur simultaneously. 



It will be found that their sum is a maximum when pt = and is then 

 0'28, the circular measure of 16 ; thus lying well below the limit of 20 

 suggested in the earlier part of this paper. The body-path, on integra- 

 tion, and taking Z = and dZ/dt initially, is Z = - pg/p 2 (1 - cospt) 

 and the amplitude is 0'0125 foot. The body, therefore, pursues 

 a simple harmonic path, whose average level is about 5/32 inch 

 below the mean level of the wing-path, and whose total rise and fall 

 from crest to hollow is almost 5/16 inch. The relative motion of wing 

 and body does not sensibly differ from that of the wing alone. 



The investigation of the question as to hovering is far less satis- 

 factory. We might assume, for instance, that the centre of pressure of 

 the wings followed a circular or elliptical orbit, the plane of the wings 

 making a small angle with the tangent to the orbit, and being variable 

 as before, so as to maintain, on the whole, an upward pressure. If we 

 make this angle a = msin pt, and form the equation d 2 Z/dt 2 = g(F - 1) 



