STUDIES IN STATISTICAL REPRESENTATION. 91 



_ — a 4 sin 4a 4 



a — »4 cos 4<* 4 . v3. 



a 77- 



i|jiHaving obtained a 4 , we have 



(28)...a 4 = - Q^( r 2 + s* 1 + fs + f n) cosec4<* 4 . 



To get the terms a 3 and « 3 we form the sum n + r 6 + r 9 and 

 the four possible similar ones. All the terms cancel each 



other except al and a 6 and we thus obtain 



a n 



(29)... n + n + n =— [a 3 .v2 sin 3(a 3 + 15°) + a 6 sin 6(a 6 + 15°)] 



17 = Mi, say 



(29a)...r 2 + r 6 + no=— [<W2 sin 3(a 3 + 45°) + a 6 sin 6(a 6 + 45°)] 



= M 2 



(29b)...r 3 + r 7 + rii=— [a 3 .v2 sin 3(a 3 + 75°) + a 6 sin 6(a 6 + 75 )] 



A = ^ 3 



(29c)...r 4 + r 8 + ri2=--[a 3 .v2sin3(a 3 + 105 o ) + o 6 sin6(a 6 +105°)] 



= M 4 

 from which the following are derived, viz., 



™ — a 3 . 2 sin 3« 



(30)... ^±^ = * = - ta n 3a 3 



) a ' M. + M, _6 as _ 2cos3a8 



TV 



(31)... and a* = ^ [> 2 + n + n ) + (n + n + n)] sec 3a 3 

 To obtain a 2 and a 2 we proceed as follows 



(32) rf-f 4 + iv -fio= — [2a 2 sin2(a 2 + 15°) + 



| a 6 sin 6 (a 6 + 15°)] = N l say, 

 (32a)... r 2 -r 5 + r 8 -r n = J*. [2 a 2 sin 2(a 2 + 45°) + 



7T 



"3 



4 a 6 sin 6 K + 45°)] = JV 2 ; 

 (32b)...r 3 -n + r 9 -r 12 = 1 r 2 a , s i n 2(« 2 + 75°) + 



thus T a6 Sin 6 ( a e + 75 °)] = ^ 



— 2 « 2 . ^3 sin 2a 2 

 (33)... Nl ~ N - =s = A tan 2o * 



r 



