PAPER BY PROF. HAGEN. 25 



From this there results as the most probable values 



z=- 0.171 

 r= + 66.199 



' If these constants are introduced into the expression for 0, the latter 

 assumes the values given the column headed A, whose departures from 

 the observed G are given in the last column. 



The surfaces of these disks measured very accurately 16 square 

 inches, aud the distance of the center of pressure was 96.500 inches. 

 After reduction to the adopted normal density of the air the con- 

 stants r for the two series of observations became respectively 



66.65 and 66.373 

 whence 



k = 2.5286 and k = 2.5178 respectively. 



The constant coefficient of the square of the velocity resulted there- 

 fore in this case as great as the series of observations III and IV would 

 have led us to conclude would have been found for square disks of 

 about 7 inches on each side; consequently the suspicion arises that 

 the increase in the value of k is not proportional to any linear dimen- 

 sion, but to the circumference of the disk. A simple consideration 

 leads to the same result. 



All previously given observations show that a disk of an area F 

 moving with a velocity c through the air in. a direction normal to its 

 plane experiences a resistance 



B = k Fc 2 . 

 If we analyze k into two terms 



k = a-\- p /3 



where j) expresses the circumference of the disk, then the first part of 

 D, namely, a Fc 2 , corresp onds to the ordinary assumption. The second 

 part 



p Fc 2 /3 = Fc. p. c. j3 

 contains, as a factor, the mass of the passing air, which is proportional 

 to Fc, also p, or the circumference of the disk, which the air touches, 

 and finally the velocity c, under which this contact takes place. It 

 appears therefore that the cause of the increase of the resistance can 

 be none other than the friction of the air against the edge of the disk. 

 However, as the experiments already mentioned in the preface have 

 shown, the air immediately adjacent to the edge of the disk flows 

 perfectly regularly past it, without taking up any whirling motion, 

 which latter first forms behind where the air protected by the obstacle 

 is touched. Friction is therefore (in accordance with the experience* 

 with water) proportional to the first power of the velocity. 



Before I computed the appropriate constants by the combination of 

 all of the observations, I made an attempt to compare among them- 



* "On the Influence of the Temperature on the Movement of Water in Tubes." 

 Hagen, Math. Abh. Akad. Wiss. Berlin, 1854, p. 69. 



