FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 109 



of a certain size, and having a given velocity in a certain fluid to appa- 

 ratus of another size and another velocity moving in a mass of another 

 fluid of a similar geometrical form. The extension of the results 

 analytically obtained for an incompressible liquid to gases led to a 

 series of interesting applications. Thus Helmholtz found, among other 

 things, that there is a limit to the size of birds beyond which the mus- 

 cles would require to do more work in proportion to the mass than 

 now. In the great vulture nature has probably reached the limit in 

 size of a creature which shall be able to sustain itself for a long time 

 in air, and thus man can have no hope by use of tbe most suitable 

 wing-like mechanism, which he could move by muscular effort, to raise 

 and sustain his weight in air. In applying the principle of comparison 

 above mentioned to the construction of air balloons and ships the inter- 

 esting result was reached that when the balloon weighs about half as 

 much as the person propelling it the relation between working force 

 and weight would be about the same as we are accustomed to see in 

 steam war vessels. 



In the years 1888-1890 Helmholtz extended his investigations of the 

 motion of fluids in order to show how in masses of air discontinuous 

 surfaces may result from the continuous action of forces. In these 

 researches "on atmospheric movements" and "the energy of the waves 

 and the winds" the inner friction of the fluids were taken account of, 

 as in his former hydrodynamic investigations. After showing, by an 

 exact mathematical treatment, that the effect of the friction at the 

 earth's surface was very inconsiderable in the higher layers of air, that 

 dissipation of the kinetic energy by friction was accomplished chiefly 

 at the surface of the earth and at the surfaces of separation caused by 

 rotary motions; and, further, that heat transference, excepting in the 

 vicinity of the earth's surface and at the inner discontinuities, is effected 

 only by radiation and by the convection of the warmer air particles, he 

 raised the question why the circulation of the atmosphere is not accom- 

 panied by higher winds than actually do occur. He assigned as the 

 reason that different layers of air are mixed by the vortices caused by 

 the rolling together of discontinuous surfaces. In this way the layers 

 become broken up and receive such a great extension of their surfaces 

 that the transference of heat and the equalization of their motion 

 through friction is much facilitated. A still more important cause of 

 the breaking up and intermixture of the various layers of air is found 

 in the regular march of waves through the atmosphere, which, as on 

 water surfaces, is caused by the superposition of two layers of air of 

 different specific gravities. The existence of such layers is only visible 

 when the humidity of the under layer becomes so great that mists 

 gather on the crests of the waves where the pressure is less, and then 

 there appear strips of parallel clouds which extend over great regions 

 of the sky. Helmholtz, therefore, was convinced that it was of the 

 greatest importance to work out the theory of waves on the boundary 



