32 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



2.026 Transformation from one to another oblique system. 



If n xy n y , n z are the normals to the planes yz, zx, xy and n x ', n y 'j n z ' the 

 normals to the planes y'z', zV, x'y', 



x cos xn x = x f cos x'n x -f y' cos y'n x + z f cos z'n x . 

 y cos yn y = x f cos x'n y + y' cos y y + z' cos z' y . 

 3 cos zn z = x' cos a's + y' cos / + 2' cos z'n z . 



x' cos s'w,' =. x cos m x ' + y cos yn x f + z cos zw x '. 

 y' cos y'fly' = s cos xn v ' + y cos ;y y ' + z cos zy'. 

 2' cos z'n z ' = # cos ^w z ' + y cos yw/ + z cos zn z '. 



2.030 Transformation from rectangular to spherical polar coordinates. 



r, the radius vector to a point makes an angle 6 with the z-axis, the projection 

 of r on the x-y plane makes an angle <j> with the #-axis. 



x = r sin 6 cos <f> r 2 = x 2 + f + z 2 



y = r sin sin <j> 6 = cos" 1 == 



V* 2 + y 2 + z 2 



z = r cos t y 



$ = tan- 1 

 x 



2.031 Transformation from rectangular to cylindrical coordinates. 



p, the perpendicular from the z-axis to a point makes an angle 6 with the 

 x-z plane. 



x = p cos 6 p = V# 2 + y 2 



y = p sin = tan" 1 - 



# 



2 = Z 



2.032 Curvilinear coordinates in general. 

 See 4.0 



2.040 Eulerian Angles. 



Oxyz and Ox'y'z' are two systems of rectangular axes with the same origin 0. 

 OK is perpendicular to the plane zOz' drawn so that if Oz is vertical, and the 

 projection of Oz' perpendicular to Oz is directed to the south, then OK is directed 

 to the east. 



Angles z^Oz = 0, 



53^ = 0, 



yVK = \[/. 



