Chap. 7] 



GRAVITATIONAL METHODS 



173 



It will be convenient to introduce the following simplified notation for 

 the components and their gradients in the xy as well as the x'y' system: 



,2 



dU 



dx 



dU 

 dy 



s Y 



dU 



= a; 



— n = a 



dx^ 



dx' 

 dU 



dy' . 



= X' 

 Y' 



d'JJ 

 dy' 



= b; 



QiJJ 



- -/2 m b' ^ (7^86) 

 dy 



= c; 



dx' dy' 



= C 



(7-48c) 



dxdy 



Hence, eq. (7-48a) may be written 



X = SLX -\- cy 



Y = ex + by. 



The following relations, as obtained from. Fig. 7-57 may be substituted 

 in eq. (7-48c): 



X = Z' cos X — Y' sin X x = x' cos \ — y' sin X 



7 = X' sin X + Y' cos X 



and 



?/ = x' sin X + y' cos X 



Divide the resulting equations by sin X and cos X, respectively, and form 

 two new equations by adding and subtracting them. When divided by 

 (cotan X + tan X) the new equations have the form 



X' = x' {u} + y' {v} 

 Y' = x' {v} + y' {w} 

 The coefficients are: 



u = a cos^ X + b sin^ X + c sin 2X; 



b - a . 



(7-48d) 



V = 



sin 2X + c cos 2X ; 



w = b cos'^ X 4- a sin'' X — c sin 2X, 

 so that eq. (7-48d) becomes 



X' = a;'(a cos^ X + b sin^ X + c sin 2X) + y' \-^ sin 2X + c cos 2X j 



X' = x' \-^ sin 2X + c cos 2X j + ?/'(b cos'' X + a sin^ X - c sin 2X) 



(7-48e) 



