130 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



/* 



2 / cos a? my (m) d> (m) dm 



/ 



T f* CO /*CO _ 



= / / y(fJL)f(ir)(cosfjiiraf + cosu.v + af)dlJ.da- 



TTi/ o *x o 



/CO /^ 



dvcosfj.*- (fa! + -a+fd *}=! da-cosfj. -a-f(cf + ) 

 U - CO 



x>CO 



=/ rfar COS //(')/() 

 i/ co 



r<o i r r-to 



2/ costfm'y(m)(f)(m)dm= I I cosfJ.(-sraf)y(fjC)f(tr)dvdfJ. 



\J vTV/ SQ\J Q 



69. In applying this theorem to the question before us, we take equations (8), 

 which will coincide with our theorem if 2 be omitted, and 



m b m b + Zh. 



e +e ) 



. . 

 v ( m) = 



d m (e+e-) 



gmb _ m (b+2 h) 



It should have been observed, that c'l t =ffm mb _ m jr+ijh' 



e +e 



Now in the case before us, /(a 7 ) is a function which is constant =bg, be- 

 tween the limits a! 0, '=a, and zero for every other value of a'. 



/CO 

 /(sr)coS)U(a a') dv 

 -co 



I "/Wcos u(r a') d-a= - {sin/x (* of) + sin/ia'} 

 \J o [A 



i r a A" 03 



Hence </>=/ / cos //( 



TTv/ cov/ o 



. sine tdu. : (9) 





when we have integrated the expression, we must write a/ instead of a'. 



This is the complete value of corresponding to our hypothesis. 



Our next process is to obtain the values of u and v, corresponding to a given 

 canal. 



70. CASE I. Let us suppose that y, b, and h, are small quantities. Neglect- 

 ing their powers, 



sin * ' 



sn tg-< 



P 



