5 



then r 3 = ~^- 



and \ (n + nj = -^(cos 18° . b sin /? + sin 36° . i c sin 2y) 



94 G. H. KNIBBS. 



Primarily we have, 

 (47)... a = g- (q 1 + g 2 + qf 8 + q 4 + $)• 



Then denoting the differences of the means from this 

 quantity by r u etc., we have 

 (48)... qi — a = ri; q2 — a = r. 2 ; etc 



- 2 sin 36° . bsin 



5 



2-jv 

 from which we obtain 



(49) b sin B= ^ . ifri + n)-r,sinl8 B 



Further 



fi-n = ^{2bcos/3(l-sinl8°)-f 2(l + cos36°)|ccos2y| 



r,-r 4 = A.J2 b cos j8 (sin 18° + cos 36°) - 



2 (sin 18° + cos 36°) \ c cos 2y I 

 Consequently ' 



(50) *L i r i ~ r » + ^ - n 

 v / '" 5 (1 + cos 36° sin 18° + cos 36° 



2 b cos /3 



1-sin 18° 



1+cos 36 c 



b cos P (2 



1 + sin 18° ' 



which determines b cos fi and then from (49) we can obtain 

 tan P and thence b. 



9. Tabulation of terms as far as 6x— The table hereunder 

 gives the trigonometrical coefficients for terms up to 6x 

 inclusive. In order to find, as is necessary, the several 

 terms in the expansion of r n , r n having the value assigned 

 to it in § 7, viz., 



6 



(51)... r n = 



a x cos (x+aj) - \ a 2 cos 2 (x+a 2 ) - 



(ri-l)7T 



We may note that, ignoring the multipliers a u a 2 ... and the 

 constant — , the co-efficients of cos <*i, sin «i: cos « 2| sin <* 2 : 



7T 



cos a„, sin a 3 ...: are given as far as the term involving 6x in 

 the table hereunder. 



