72 REPORTS ON THE STATE OP SCIENCE. 



The column of E and F gives 



E (r°) = zn /K = E^ - /E, 



F (r °) = E (90 - r) = zn (1 - /) K 

 and 



f = am /K, /K = 2ty. 



If the small Theta function is preferred, K must be replaced by \v, 

 and ®/K by 6 {\irj), H/K by 0, {\irj), but the numerical entry in the Table 

 is unaltered. 



The value of A, B, C, D is chosen as the quotient of two Theta f imctions, 

 because it can be expressed algebraically, and calculated to any desired 

 number of decimal places, for a Division value of the Elliptic Function ; 

 and so the accuracy may be tested of the terms of a series employed in the 

 computation. 



Thus at the Bisection, / = |,r= 45, and 



E (45) =F(45) = zn AR = | (1 - «?) 



k =s sin 6, k = cos 6 



V 



D (45) = C (45) = (*±£V 

 A(46) = B(4B)=(^)i 



Bn * K =V / (lT?)- m4K = V / G ;'«)• to » K =^ 



At Trisection, in the region 3> 6> 1, 



k2 _(6+1) 3 (3 - 6) ,,_ (6-lf (6+3) 



166 ' ~ "166 . .. 



D(30) = C(60) = @— ^* 



D(60) = C(30) = ^|jJ 



A(30) = B(60) = !^ 

 A(60) = B(30) = 2>/Z.p^- )t 



E(30) = F(60) = 9 -^ 



E (60) = F (30) = ( i=- 6) ^ 

 6 



6=3! N/6-fV2 =2 . M2 

 2 



The value 



makes 6 = 45°, and the Table has been checked by these values. 

 Also 6 = s/3 makes 6 = 75°. 



