424 Prof. M. F. FitzGerald. 



,727 



^= = 0(P_ - 1), the bird being 1 Ib. weight, 



AAVX1 - ft coapf) - 1 } 



and, on integration, all terms containing t 2 or t as factors must vanish, 

 in order for the motion to be, on an average, horizontal. That con- 

 taining f as a factor will not vanish unless 



2&AV 2 m = 1 which fixes m, ........................ (B) 



and gives = ^ ain.pt as there is no term with t as factor in Z. 



By differentiation = pS cos (pt + 0), 

 Cut 



whence _^-(Z - z) = - / ^L sinpt +pS cos (pt + 6)\ 



at [^ p j 



Inserting their values in (A) for the quantities involved, we get 



P*L* y 



jo 2hkV~^ 1 ~ fJ ' GOSpt ^ dt 



r 2iT 'p rug ~\ 



= (1 - {Ji cospt) -- sinpt +pS cos (pt + 0) dt. 

 Jo L P J 



On integration, the only terms that do not vanish give the equation 



which is a quadratic for ^ namely, 



/* 2 -2&AVpScos0/x + 2 = ..................... (C) 



Taking the integral of one side alone we find 



w = /ffS cos , D . 



2 



Eliminating ft from these we get 



If now we draw a curve whose ordinate, y, is j^S cos 0, and abscissa, 

 x, is W, we find it to be of the form shown as AB or CD in fig. 1, with 

 a symmetrical branch below the axis of x, not shown in the figure. 

 On account of the steepness of the first part of the curve CD, that 



