180 



GRAVITATIONAL METHODS 



[Chap. 7 



(5) (2 + 4) = a;' + a'/ = -2a"C/^ - h"Uy^ 



(6) (14-3) = A2 + A3 = -2a'C/^ + h'Uy. 



(7) (1 -3) = A2 - a; = -V3a'l7A + \/3b'C/„ 



(8) (2-4) = a',' - a',' = - V3 Si"U^ - V3 b"t/«. 

 Multiplying (5) by a' and (6) by — a" and adding, 



Uy, = — - — [(A^ + A3') - - (a;' + ad] . 



a'b" + a"b'L '^ '' a" J 



Multiplying the same equations by b' and b" respectively, 



2U^ = ^ [(A^' + A^) + ^ (a;' + A^')] . 



a"b'4-a'b"L «^ ^ b" J 



Multiplying (7) by b" and (8) by b' and adding, 



t/A = 



(A2 - A3') + ^, (a;' - A 



V3 (a'b" + a"b') 

 Multiplying (7) by a" and (8) by —a' and adding, 



.] 



(7-52c) 



{7-52d) 



(7-52e) 



t/« = 



V3(a 



With the notations 



a" 

 = 



-^ [(Az - A3') - -^ (a;' - a;')1 . (7-52/) 



"b' + a'b'OL a"' ''J ■" 



VS (a"b' + a'b") 

 V3 (a"b' + a'b") 

 a" 



P = 



r = 



a'b"+ a"b 

 b 



TTIT/ (P = 0\/3) 



a'b" + a"b 



TTiT, (r = qVs) 



(7-52^) 



b" 



formulas (7-52c) to (7-52/) may be written 



c/« = o[(A^ - aD - s(a;' - a;')] 



Uy, = p[(A2' - A3') - s(a;' - A^')] 



ua = -q[(A2' - A3') + t(A;' - a;')] 



2U.y = -r[(A2 + A3) + t(A;' + A^')]. 



> {7-52h) 



In a similar manner, the following expressions are obtained for positions 

 (1) and (2): 



