584 



Mr. H. Bateman on the Velocity of an 



To apply these results to a practical case we must endeavour 

 to obtain an approximate representation of the results of: 

 observation by means of some linear combination of" these 

 particular forms of / (a). The constant /3 which determines 

 the velocity at the surface is found from the value of a 

 which makes / (a) vanish. 



Table of Solutions. 



/('.)• 



1 dx 

 x di} 



». 



—1 n 



cos l -^ 



/3 



1 



>y2_ a 2 



1 



1 



X 



a tan- (/x V^ 2 — a 1 ) 



a cos 1 , 



V« 2 +^^ 







9(1 +/*V) 



1 



u si a* — a. 2 



1 V\+/ji2a2 -pWa' 2 

 s — tan— 1 / :.-- — r 



f(|3-«) 





J?V / l-f-4«2^2 



2j* //32 



juC^-a") 



The solution given by Prof. Knott is fourth on the list, 

 the fifth solution corresponds with the law of velocity con- 

 sidered by Wiechert and Zceppritz. The other solutions by 

 themselves give irrelevant laws of* velocity, but by combining 

 them with the others we may obtain possible laws of variation 

 of the velocity. 



If there is any difficult v in expressing the results of obser- 

 vation by means of the function given in the table, it may be 

 convenient to assume different expressions for S for different 

 ranges of values of 0: for instance, we may assume 



S = a + &0-r-c0 2 for 0<<9<<£, 



S = a' + b'9 + c'6 2 for <£<0<tt, 



provided the constants are such that the condition of con- 

 tinuity 



a + b<j> + c$* = a! + b ! (f> + c'<t> 2 

 is satisfied. 



