Manchester Memoirs, Vol. xliv. (1899), ^'■'- ♦»• 



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We now assume that i) and i' arc pcritxlic, for 

 example that 



(j = (Jo + (^1 cos// (41) , 



v/u = ftcos{/>t-i-t) (42), 



where the periodic time r is related to/ according to 



T=27r/p. 



At this stage the criticism may present itself that the 

 assumed motion involves a reaction for which we have 

 made no provision. In practice the reaction is supplied 

 by the inertia of the body of the bird to which the wings 

 are attached. The difficulty would be got over by 

 supposing that there are several planes executing similar 

 movements, but in different phases regularly disposed. 

 It seems hardly worth while to complicate the present 

 investigation by introducing a vertical movement of the 

 weight. 



By (40) the whole pressure at time /, perpendicular to 

 the plane is 



k-i"?/-^ {^0 + ^*1 008// + /) cos (/>/ + €)} , . . (43) 



Of this the mean value is to be equated to the weight IV 

 supported, so that 



Jl''=i.S /rOo (44). 



The horizontal component of the whole pressure at time t 

 is 



S.ur.{d + rl;^]6 (45). 



and of this the mean value is to be supposed to be zero, in 

 order that the plane may move with uniform horizontal 

 velocity. Thus 



eo' + 5(^i' + |/3 9i cose = .... (46). 

 Again, if JF6'^ be the (mean) rate of expenditure of work, 



JFC/= S Kir.f{d + vji()vd{fJT) - 6; Uir\i'j d, cos t + I'r) . (47 ). 



