ON THE CALCULATION OF MATHEMATICAL TABLES. 55 



Lemniscatc 5 section: r=18, 86. 



5. The formulas to be employed are given on p. 7, Eeport 1911, where 

 we put 



_ (c-l)(o2_4 c-i) _ (c-H)(c^-4 c-l) 



^>- (c + l)3 '«2- (c_l)3 



(c + l)Mc-I). (c + lH(c-l)^ 



"'- 2cJ '''^- 2ci 



_ _ (c + l>V(c-iy.(c^-4c- l)i 



, _, va _(c + m(c-l)X(c^-4c-l)i 



taken from Phil. Trans. 1904, p. 264 ; and apply the numerical values of 

 c in the three regions, I, II, III. And with 



u _(c + 3)(c2-4c-l) , _(3c-l)(c2-4c-l) 



Oj — ^ ,02 :j 



20ci{c-^-+l)i 20c?(c- +1)' 



c c 



4c*(c-^ + I)i 4(-- + l)i 



c c 



4ri(c-l^j), 4(0-1 + 1)1 



c c 



Begion II. 



C(36)=:^, D(18)=^^; , A(36)=4' , B(18)=4' 



c— =4 sin 54, c + -=4 sin 72 

 c c 



c=2 sin 72 + 2 sin 54=4 sin 63 sin 81 



c--^-4=-8sin218, c-i + l=8sin354=(2sin54)3 

 c 



c + ^— 2=4 sin 72 + 4 sin 30=8 sin 51 sin 69 

 c 



c + ^— 2=4 sin 72—4 sin 30=8 sin 39 sin 21 

 c 



,. 2 sin^51 sin-'69 sin254 ,(, oo • eo • oi • %on ■ %c-i ■ ■>ka 



c, "= . „„ . - ^ , c,^°=32 sm 63 sm 81 sm'39 sm321 sm^S^ 



' sm 63 sm 81 



^ ,0^2 sini54 sin5l8 sin239 sin221 

 sin 45 sin 63 sin 81 



g io_32jinj54 sinn8 sin^S l 8in^69 sin 63 sin 81 



sin^ 45 



I 2 



