106 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



/ ii a^4 2 z> a^4 v 



curl u A = v ~r~ ~ ~^~' 



v u-\- V dv u -\- 1) dz 



tt t .B 



CUrl v A = ; \/ ; -T > 



u + v dz i u + v ou 



5.40 Helical Coordinates. (Nicholson, Phil. Mag. 19, 77, 1910.) 



A cylinder of any cross-section is wound on a circular cylinder in the form of 



a helix of angle a. a = radius of circular cylinder on which the central line of 



the normal cross-sections of the helical cylinder lies. The z-axis is along the 



axis of the cylinder of radius a. 



u = p and z; = $ are the polar coordinates in the plane of any normal section 



of the helical cylinder. </> is measured from a line perpendicular to z and to the 



tangent to the cylinder. 



w = = the twist in a plane perpendicular to z of the radius in that plane 



measured from a line parallel to the #-axis: 



x = (a + p cos </>) cos 6 + p sin a sin 6 sin (/>, 



i. 



y = (a + p cos <) sin 6 - p sin a cos 6 sin </>, 

 z = a ff tan a. + p cos a. sin $. 



/?!=!, ki = ~> 



p 



a 2 sec 2 a + 20p cos + p' 2 (cos 2 </> + sin 2 a, sin 2 c/>) 



5.50 Surfaces of Revolution, 

 z-axis = axis of revolution. 

 p, 6 = polar coordinates in any plane perpendicular to z-axis. 



1. ds 2 = dz 2 + dp 2 + p 2 d6 2 



du 2 dip dw 2 

 = W + Jtf + hf' 



In any meridian plane, z, p, determine u, v, from: 



2. /(z -f Z*p) = W + MJ. 

 3- W = 0. 



Then u, v, 6 will form a system of orthogonal curvilinear coordinates. 



