86 U. 8. COAST AND GEODETIC SURVEY 
A sin a’ cos a=A’ sin a’ cos a’— 2D F, cos {4(b—a)7+ B} sin a’ (381) 
A cos a’ sin a=A’ sin a’ cos a’— 2 Fy, sin {4(b—a)r+ 8} cos a’ (382) 
Subtracting (382) from (381) 
A sin (e’—a)=2z F, sin {$(b—a)r+ B—a’} (383) 
Multiplying (379) and (380) by cos a’ and sin a’, respectively, 
A cos a’ cos a=A’ cos’ a’ —z F, cos {4(b—a)7+ B} cos a’ (384) 
A sin a’ sin a=A’ sin? a’—z F, sin {3(b—a)7+ B} sin a’ (385) 
Taking the sum of (384) and (385) 
A cos (a’—a)=A’—2z F, cos {4(b—a)t+ B—a0’} (386) 
Dividing (383) by (886) 
> F, sin {4(6—a)7+ B—a’} 
— F, cos {4(b—a)r+ B—a’} 
tan (e!—a)=F 
From (386) 
(387) 
A’—® F, cos {4(b—a)r+ B—a’} 
cos (a’—a) 
249. Substituting the value /, from (376) and the equivalents 
f(A), R(A), fy (CB) at CA) GCA) and 191023) for vA A, B, a’, a; and 
B, respectively, we have by (387) and (388) 
tan [¢(A)—¢/(A)]= 
sll sin 3(b—a)r 
© ¥b—a7 2B) sin {4(6-a)7—5(B)+4'(A)} 
B'(A)— 3 ah —ayr PB) 008 {46—a)r—0(B) +8'(A)} 
A= 
(388) 
180 sin 4(b—a)r (389) 
Ria) — Se sn OO FRB) cos (4(b—a)7—1(B) +1"(A)} 
cos [¢ (A) —¢’(A)] 
R(A)= 
(390) 
250. Formula (389) gives an expression for obtaining the difference 
to be applied to the uneliminated ¢’(A) in order to obtain the true 
¢(A), and formula (390) gives an expression for obtaining the true 
amplitude R(A). These formulas cannot, however, be rigorously 
applied, because the true values of R(B) and ¢(B) of the disturbing 
constituents are, in general, not known, but very satisfactory results 
may be obtained by using the approximate values of R(B) and ¢(B) 
derived from the analysis or by inference. 
By a series of successive approximations, using each time in the 
formulas the newly climinated values for the disturbing constituents, 
any desired degree of refinement may be obtained, but the first 
approximation is usually sufficient and all that is justified because 
of the greater irregularities existing from other causes. 
251. Form 245 (fig. 19) provides for the computations necessary in 
applying formulas (389) and (390). In these formulas the factors 
180 sin 4(b—a)r 
represented by (b= aia 
and the angles represented by 
