SnOKT MEMOIRS ON METEOROLOGICAL SUBJECTS. 437 



case of the absence of every friction al resistance, this entire force will be 

 applied to the acceleration or retardation of the rotatory movement of 

 the particles of air according as these are respectively approaching to 

 or receding from the center. When, however, any friction exists, this 

 force will be almost exclusively used to overcome this frictional resist- 

 ance. If, now, we designate this frictional resistance by F'^ which acts 

 in the direction perpendicular to the radius along the portions cos » 

 of the path, as we have previously designated by F the resistance 



acting in the direction of the radius along the portion — , then the 



following relations hold good : 



F' is very nearly = 2{n sin c + u] 



![ _ \iuj . 



F' ~ V cos i ' 

 2 11 sin <p + 2 u 





F = 



V cos I 



m 



This latter expression is always positive, and attains its greatest value 

 in the neighborhood of the earth's surface, where the frictional resist- 

 ance is the greatest. With increasing altitude, where only swiftly mov- 

 ing air rubs against more slowly moving air, the friction rapidly dimin- 

 ishes, w^hile the angle i also approximates to its limiting value of zero. 



If, now, we substitute this value of F in the equation (F), after we 

 have previously multiplied numerator and denominator by v cos i, we can, 

 with slight error, eliminate the common factor v cos i (2 w sin ^ -f- w), 

 wherebj' the equation acquires the following form : 



{ (- 



-Ji' = .TU^^-T • rrr (2 H siu <f -f 2l) V COS * < 1 -I- -^— 



Cut since, according to the definition of i, the ratio ~: ycos* = tangi, 



at 



therefore the definitive expression for the gradient in the case of a 



spiral movement of the air is as follows : 



(11) Ap-J_ •? (2»sin^ + «)i) 



"~ 287.4: ■ T ' cos i 



in which 



V cos i 



u = 



r 



In reference to this equation (II), we must make the following remarks: 

 First, that part of the gradient that results from the term 2 wiJsin 99 is 

 really entirely independent of the angle i or of the direction of the currents 

 of moving air, but the formula allows it to increase with increasing values 

 of i, which must be incorrect. Secondly, the second term which repre- 



