484 



traKsactIoiJs of section a. 



For a cyclouic Anemoid, a is the angle of incurvature. 



For an anticyclouic Anemoid, change a into —a; a then becomes the angle at 

 excurvature. 



Make the following substitutions : 



cj) — a =■(()' 

 ^ = sin a/-v/(cos' a-M^), - tan h 0' = (cos a-n)i{cos a + /a)~5 cot^\lr 

 y = sin aj^{fi" — cos- a), — tan ^ 0' = (/Li-cosa}* (jLi + cosa)~icothJx 

 ^ =Pi = 9X- 



Also suppose that when <^ = 0^, x = h^y = k. 

 The equations of the Anemoid are shown to be 



y/A =fi^)l/(i>o) (-r - m = IMzli^ 



where 



f(<f))= - cos(j>' e~'l{cosa~iJ.co3 (f)') 

 ^ (</>) = ( ~ M + M' cos a cos 0' + sin a sin (i>')e~'j { (1 - /*•) sin a (cos a - /x cos ^ ') } 



;>, ^//•, or y, ;^, being used according as /x < or > cos a. 

 When IX = cos a, the general equations do not apply. 



Turn the axes through an angle a about the point where the air-particle falls 

 into the storm-centre. The equations may be written : 



.r/a= - (Z' + l-tan'^a)e-2 «//a = 2Z tanae-z 



where 



Z = l-tana cot ^ <^' 



The case /x =» 1 is also exceptional. In this case let 



Z = sin^ (</)-2a)/3in A0 



and suppose that when Z = Z^, .r = h,y = k. The equations are 



y/A=/(Z)//(Zo) 



where 



sina(.r-A#={F(Z)-F(Z,)l//(Z„) 



/(Z) = 1/Z - i (cos a)/Z^ - i cos a 



F (Z) = (cos a)/Z - Kcos 2a)/Z'' + J log Z 



If /iisO, the Anemoid is an equiangular spiral, which becomes a circle when 

 <i = 0. 



If a = 0, there is neither incurvature nor excurvaturej and the path may be called 

 a right Anemoid. The equations are : 



,X<1 



iu>l 



{l-ix)ylc = -;x + C0Si/a 

 (1 — /x) \/(l —ix'^')xjc = fi^ — sin yjr 



9e.r' = (c~y) {ic + yY 

 (f*-l)y/c = /x-cosh X 

 (m-1)\/(m''-1)''*/c= -/xxfsiDh;^ 



assuming that .r = when ^/' = 0, x = 0, ?/ = c. 



