(i) 



(50) 



we find from (2) and (3) : 



H^ sm ).^ = — qA cos Cj -|- aB sin C^ -j- kA sin C^ ■ 

 H^ cos P.J = qA sin C^ -\- aB cos C^ -\- kA cos C^ I 



H^ sin P.J = — qB cos C^ — ciA sin C^ -\- kB sin C^ I 

 H^ cos ^j = qBsin Cj — aA cos C^ -\- kB cos C^ 



The quantities H and / can then easily be calculated by means 

 of the following? relatively simpb formulae, provided the wind 

 variation and friction coefficient be known: 



H^ sin (/j — C'l) := — qA — aB sin Ex \ 

 H, cos {)., — C,) = kA-\- aB cos L I 



H, sin (;., — C,)= — qB — a A sin h i ' ' ' ' ^""^ 

 H^ cos (;., — C,) = kB — aA cos A | 



and further : 



IJ^"- _ H^"- = {k' + q' — a') {A' — 5=) -f 4 ka AB cos A ) 



H,' f II]' = {k' ^ q' ^ a^) {A' + i?^) + 4 qa AB sin L \ ' ^^^ 



Form. (3) can l)e represented by an ellipse, the radius vector 



being the resultant velocity, and the great axis forming with the 



y, V (North) direction an angle a determined by the expression : 



2 AB cos h 

 tan<i '1 a ^= —- — , (7) 



Likewise the gradient vector can, according to magnitude and 

 direction, be represented by an ellij)se, its great axis making with 

 the North direction an angle «', determined by the form. 



, 2 H, H, cos {X-K) 

 ^^''9 - « = jT'^_H^ ^^ 



From (4) follows : 



H, H. sin {?., - A J = {k' +q' + a-) AB sin A + qa {A' + B') ] 



H, H, cos (/, — /J = (F +q'~ a') AB cos A — ka {A' - B')\ ^- ^ 



If then we put: 



2ak 



tan(i2m=z :■ -, (10) 



F +5 — « 



tang 2 «' = tana (2« — 2m) (II) 



m :=z a — «'. 



Although, therefore, for the case of a periodical!}- varying gradient, 



the angle of deviation between gradient and wind direction does 



not assume a simple form and is a rather complicated function of 



the time, there is (if we admit the form. (1) as suitable for the 



