PAPER BY PROF. HELMHOLTZ. 79 



movemeut takes place in an analogous manner, only slower. When 

 this is not the case and when the friction retains its value unchanged 

 then will the influence of the friction on the increased mass be very 

 much less than upon the smaller mass. In consequence of this the 

 greater mass will show the effects of its inertia as influenced much less 

 by friction. 



It is to be remarked that the potential P remains unchanged by the 



iP 1 



increase of the mass, but the force ' is reduced to - of its value and 



,U' n 



that the whole process as alreatly remarked requires for its completion 

 n times the time. 



Since the density and pressure are to remain unchanged therefore 

 also any temperature difterences that are present retain their magnitude 

 and influence and do not disturb the relations implied in the mechani- 

 cal similarity. 



Unfortunately we can not imitate in small models the varying density 

 of the atmosphere at different altitudes since we can not correspond- 

 ingly change the force of gravity that is included in the expression 



■)P 



^—- . Our mechanical comparisons are only able to imitate an atmos- 

 phere of constant density. Such an one must, as is well known, have an 

 altitude of 8026 metres at 0^ C. in order to produce the mean baro- 

 metrical reading of 76 centimetres of mercury. If we desire in a model 

 to represent the atmosphere by a layer of one metre in altitude, then 

 we would need to reduce the day to 10. S seconds, or the year to 65. 5 

 minutes, and the influence of friction in movements at velocities that 

 correspond to those of the atmosphere would in a small model be 8026 

 times as great as in the atmosphere. The loss of living force in the 

 atmosphere during a year would therefore correspond to that lost in 



65 5 

 our model in -^ — of a minute, which corresponds to less than a half a 

 8026 



second. 



On the other hand it is possible with the measured value of the 

 friction constant of the air to compute for some simple cases how long 

 a time would be required in order to reduce to one-half of its velocity 

 any motion that is hindered only by internal friction. In this case the 

 assumption of a constant density is for our purpose more unfavorable 

 than the adoption of the actual variable density. 



Assume that a stratum of air whose constant density is such as that 

 of the lower stratum of the atmosphere, spreads over an unlimited plane 

 and has a forward movemeut whose velocity is u in the direction of x 

 parallel to the plane. Let z be the vertical coordinate, then the equa- 

 tion of motion for the interior of the mass is 





,3 = ,.„„,.. . (2) 



