the Goldsdimidt 2)yndmo. 767 



pM d , . - x d , . n 



A 's= -^-j£ ( Sln * • Z/i) =cic 2 c 3 ?/ ^ (sm . cos 2(9) 



= C1C2C3//0 • 2 ^g (sin 30 — sin 0) 



— \c\W$y§ (3 cos 30 — cos 0). 



LeUis ignore c v c 2 c s . . .y for the present. They evidently 

 come in as multiples only. 



1 d 

 y± = 2~TQ ( sm # • 3 cos 3(9 — sin cos 0) 



= |^(3 sin 40-4 sin 20) 



= j (3 x 4 cos 40 - 4 x 2 cos 20) 

 = 3 cos 40-2 cos 20. 

 Now suppose this is A 4 cos 40 — B 4 cos 20. 



#5= -^ sin 0(A 4 cos 40— B 4 cos 20) 

 = ^{A 4 sin50 — (A 4 + B 4 )sino0 + B 4 sin0} 



= |{5A 4 cos50-3(A 4 + B 4 )cos30 + B 4 cos0} 



= ^(15 cos 50-15 cos 30 + 2 cos 0). 



Call this A 5 cos 50 — B 5 cos 30 4- C 5 cos 0, and note that 

 this makes A 5 — B 5 + C 5 = A 4 — B 4 = 1. 



Similarly 



y G = i!6A 5 cos 60-4(A 5 + B 5 ) cos 40 + 2(B 5 + C 5 ) cos 20} 



= 3A 5 cos60-2(A 5 + B 5 )cos40 + (B 5 + C 5 )cos20 



= 4(45 cos 60-60 cos 40 + 17 cos 20). 

 Also 



x 7 = i{45 x 7 cos 70-105 x 5 cos 50 + 77 x 3 cos 30-17 cos 0} 



= i{315 cos 70-525 cos 50 + 231 cos 30-17 cos 0}, 

 and 



y 8 =i {315 x 8 cos 80-840 x 6 cos 60 + 756 x 4 cos 40 



-248x2 cos 20} 



«= 315 cos 80- 630 cos 60 + 378 cos 40-62 cos 20, 



with multiplier c x c 2 c s c 4 c h c 6 c 7 Cg y . 



