208 M. K. Zoppritz on some Hydro-dynamic Problems 



dependent of /, and therefore only = </)(//). These require- 

 ments are satisfied by the function 



1 M ©iu(2n + l)7r-^cos(2n+l)7r-|^ 



w 



= f2 



"«©„( 2 n + l)4 + (^ 6 of(2„ + l)4 



f 



+ b \ 



</>(V) cos (2n + l)ir ^r d\, 



J-b ^ 



in which, for brevity, the hyperbolic sine and cosine are repre- 

 sinted by ©in and gof ; so that 



©in«=i(/- e -«) ? £o{*r=±(e a + e-«y 



If the velocity of the air be taken as independent of y, and 

 consequently (j>(y) be put = w t , then becomes 



M»(-l) ©m(2n + lK^cos(2«+l>| 



7r 2n + l h (Sn+'IW^ r/0 , 1N /* ' 



@Hl(2n + l>rgj + v — 2^ Z - e °^ 2n + 1 ) 7r 2^ 



If herein we put the breadth 26 = <x> , the resulting formula 

 will be the same as that for the velocity of the stationary mo- 

 tion in an entirely unlimited sheet. The fraction under the 



symbol of summation takes for 26 = c© the form ~. By dif- 

 ferentiating both numerator and denominator according to ^-= 

 we get, after inserting 26 = oo , 



w= M.K^) 2 <^ cos(2n+1) , 



7r ph + 1 2n+l y 26 



But this sum is, for all positive values of y which are less 



7T 



than 6 (the value y = b itself is excluded), equal to j ; so that 



in reality w receives the above-found value (10). 



The motion in the surface, consequently for a?=Q, if the 

 assumption be retained that ^(y) = w 1 , is represented by 



tof ifcrlE «»B(2n + l)Tr^ 



which expression, for p = <x> , changes into w x itself. 



If this series were capable of summation, although only for 

 y = 0, it might be possible by suitable experiments to ascer- 



