du 



' u-y- a. 



ax 



506 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



Now, the moving force on the solid PN 



rz + ^4^ , ( dz \ \ rz + -oc / ^ du \ 



1 Z' 4.^ 



.We have omitted the part n^ J ^'''' dyu^ + Q^ + Q' from the circmnstance 



that it does not depend on the difference of velocity or of resistance, and, there- 

 fore, is no part of our force. 



Perhaps it would be well to call p the horizontal pressure, instead of tlie 

 whole pressure, as by this means we should have been spared the apparent in- 

 correctness of omitting a part of the results ; but I have preferred retaining the 

 above, as the more usual mode of proceeding. 



By integrating the above expression between limits, we obtain for the 

 moving force 



1 , 1 / rf^ \ 2 du 



and the mass is gaz; hence the accelerating force is 



(du\ dz du 



where -j- is the total differential coeflBcient of u with respect to t. 



By substituting for u its value 



ca . 27r ca . /^ 



-— . sin ^— ct—x or — . sin f 

 h A h 



we obtain 



27rca ^ / dx\ 2'Tra n litac 



h\ 



a I dx\ Z'TTa /J Zirac ^ 



cos o [ c — —J =g ■ —^ — . cos t> — u -^-—— cos o 

 \ dt) ^ X Xk 



C2 



by equating coefficients, we obtain t-=^ as the first part, and the other part is an 

 identity, 



7. We may vary the last part of the process in the following manner. 

 Since the moving force is 



^ pdy- r^^'" p'dy. 



