PROFESSOR IN BERLIN 297 



had been able to discover the integrals proper to the conditions 

 of the given special cases from the differential equations pro- 

 posed for the motion of liquids and gases, taking into account 

 the pressure and friction, from which these resistances could 

 be calculated. On the other hand, we have plenty of experience 

 in regard to ships of the most varied construction, since we 

 know the amount of force required in order to give the desired 

 velocity to any ship or boat, and we have succeeded in dis- 

 covering the most advantageous forms for the body of the ship, 

 and for the size and shape of the motor apparatus ; in the air, 

 on the contrary, apart from the few experiments with balloons, 

 birds are the only instances we have of flying machines. This 

 consideration led Helmholtz (by means of the general hydro- 

 dynamic equations that hold good for liquids and gases) to 

 transfer the results of experiments made with ships, to the 

 corresponding problems in aerostatics. He shows, by rigid 

 mathematical reasoning, that it is possible to transfer the empirical 

 results obtained from a fluid and from apparatus of given size 

 and velocity, to a geometrically similar mass of another fluid, and 

 to apparatus of other sizes and other speeds, and establishes 

 the ratio in which the velocities, the pressure, and the corre- 

 sponding energy must be magnified, if the ratio of the physical 

 constants of the fluids is given. 



In the application of this principle there is, indeed, the objection 

 that the density of the air alters perceptibly under pressure. 

 But since air can escape freely on all sides, and the most 

 successful results appear to be produced with the lower velo- 

 cities of wings or screws, only those differences of pressure 

 come under consideration which are caused by the accelerations 

 of the displaced particles of air, and these, with the altered 

 volume of air depending on them, may be disregarded (as 

 Helmholtz shows), so long as the resulting velocities are 

 negligible in comparison with that of sound. It follows, 

 amongst other things, that the size of birds must find its limit 

 unless the muscles could be further developed in the direction 

 of performing more work with the same mass than they do at 

 present. In the structure of the Great Condor, Nature has 

 apparently reached the limits of size at which any creature can 

 soar upward by its wings, and remain a long time in the air. 

 Man, in his opinion, has no prospect of raising his weight into 



