438 MATHEMATICAL APPENDICES 43* 



which we also might have deduced, by a much simpler calculation, 

 from the formula 



p = torNmmd* ds sin s cos 2 s e - 

 or its equivalent 

 p = 



We have obtained this formula in 35 (p. 72) in the form 





and from it have drawn conclusions respecting the momentum 

 and force of reaction of a stream of air, and also respecting the 

 resistance of air. 



The magnitude of the resistance is calculated in another way 

 in some memoirs. 1 It has been thought that the formulae for Q^ 

 and Q 2 may be interpreted as if < ^Q l represents the pressure which 

 the forward face of a body moving with speed a experiences in air, 

 while 2Q 2 represents the pressure of the air against the hinder 

 face. Then the difference of these two magnitudes 



would give the resistance per unit area, and this reduces to 

 Ca) = 



on neglect of higher powers of a. The resistance would thus 

 consist of two parts, of which one would be proportional to the 

 first, and the other to the third, power of the speed a. 



This mode of interpreting the formulae was first employed by 

 Him, 2 and the contradiction between his formula and experi- 

 ment led him to raise objections to the validity of the kinetic 

 theory, which were, however, answered by Clausius. 3 It is 

 sufficient here to point out that the deduction of the expressions 

 for Q! and Q 2 are not valid for a rigid bounding surface, but only 

 for a hypothetical plane in the interior of the gas. 



1 W. B. Smith, Zur Molecular-Kinematik, Gottingen 1879; E. 

 Toepler, Zur Ermittlung des Luftwiderstandes nach der Mnetischen 

 Theorie, Wien 1886; G. Sussloff, Journ. russ. phys.-chem. Ges. xviii. p. 79, 

 1887. 



2 Him, ' Kecherches sur la Kesistance de 1'Air en Fonction de la Tempera- 

 ture,' Mem. de VAcad. de Belgique, xliii. (2) 1882. 



3 Clausius, 'Examen des Objections faites par M. Him,' Bull, de VAcad. 

 de Belgique [3] xi. p. 173, 1886. 



