PAPER BY PROF. HELMHOLTZ. 77 



on the average 8 kilometers per bour for the whole duration of the 

 experiment, and 10^ kilometers attained by intense work. He is there- 

 fore not very far behind the limit that my computations show attaina- 

 able with a balloon of such dimensions. 



In the preceding computation we have however only taken account 

 of the ratio between the effective force and the weight, and have 

 assumed that the form of such a balloon and of its motor can be 

 attained with the materials at our disposal. But here seems to me to 

 lie one of the principal difficulties of the practical execution. For the 

 parts of a machine made of rigid bodies do not by a geometrically 

 similar increase in their linear dimensions retain the necessary stiff- 

 ness ; they must be made thicker, and therefore heavier. If on the 

 other hand with small motors one would attain the same effect, by 

 means of greater velocity, then work is dissipated. The pressure 

 against the whole surface of a motor (a ship's propeller, or oars or pad- 

 dles) increases as q 2 r. If this pressure, which determines the propel- 

 ling force, is to remain unchanged, we can only diminish the dimen- 

 sions in so far as we increase n, and therefore also the velocities ; but 

 then the work iucreases also as q 2 n r, and therefore proportionally to n. 

 Therefore one can work economically only with relatively slew-moving 

 motors of large surface. And to realize this in the necessary dimen- 

 sions without too great a load for the balloon will be one of the great- 

 est practical difficulties. 



