MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



2.020 Transformation of coordinates in three dimensions. 



2.021 Change of origin. Let x, y, z be a system of rectangular or oblique coor- 

 dinates with origin at O. Referred to x, y, z the coordinates of the new origin 

 O' are a, b, c. Then referred to a parallel system of coordinates with origin at 



O' the coordinates are x', y f , z' . 



x = x' + a 



y = y' + b 

 z = z' + c 



2.022 Transformation from one to another rectangular system. Origin un- 

 changed. The two systems are x, y, z and x' y' z'. 



Referred to x, y, z the direction cosines of x' are k, m^ n\ 

 Referred to x, y, z the direction cosines of y' are k, W2, n% 

 Referred to x, y, z the direction cosines of z' are / 3 , w 3 , n$ 



The two systems are connected by the scheme : 



ky' 



z = n\x + 

 li 2 + wi 2 + 



y = 

 ' 



n\z 



+ /2 2 + 



-f- 



~}~ 



= o 



kk 



kk 



= o 

 = o 



= o 



2.023 If the transformation from one to another rectangular system is a rotation 

 through an angle 6 about an axis which makes angles a, ft, 7 with x, y, z re- 

 spectively, 



2 cos 6 = k + ^2 + 7*3 i 



