14 U. S. GOAST AND GEODETIC SURVEY 
greatest deflection due to the moon is about 0.022”’ and the greatest 
deflection due to the sun is less than 0.009’’ of arc. 
44, To simplify the preceding formulas, the quantity involving the 
fractional exponent may be developed by Maclaurin’s theorem into a 
series arranged according to the ascending powers of r/d, this being 
a small fraction with an approximate maximum value of 0.018. Thus 
1 
{1—2(r/d) cos z+ (r/d)?}? 
=1+3 cos z (r/d) 
+3/2 (5 cos? z—1)(r/d)? 
+5/2 (7 cos? z—3 cos z)(r/d)?+ ete. (20) 
45, Substituting (20) in formulas (16) and (17) and neglecting the 
higher powers of r/d, we obtain the following formulas: 
F, /g=3 (M/E) (a/d)’ (cos’ z—1/3) (r/d) 
+3/2 (M/E) (a/d)? (5 cos? z—3 cos 2) (r/d)? (21) 
F, /g=3/2 (M/E) (a/d)? (sin 2 z) (7/d) 
+3/2 (M/E) (a/d)? sin z (5 cos? z—1) (r/d)? (22) 
46. If r, which represents the distance of the point of observation 
for the center of the earth, is replaced by the mean radius a, it will be 
noted that the first term of each of the above formulas involves the 
cube of the ratio a/d while the second term involves the fourth power 
of this quantity. This ratio is essentially the moon’s parallax ex- 
pressed in the radian unit. These terms may now be written as sepa- 
rate formulas and for convenience of identification the digits 3” and 
“4”? will be annexed to the formula symbol to represent respectively 
the terms involving the cube and fourth power of the parallax. Thus 
F,; /|g=3 (M/E) (a/d)? (cos? z—1/3) (23) 
Fy, [g=3/2 (M/E) (a/d)*(5 cos’ z—3 cos 2) (24) 
F,;, [g=3/2 (M/E) (a/d)? sin 22 (25) 
F 4 /g=3/2 (M/E) (a/d)* sin 2 (5 cos? z—1) (26) 
Formulas (23) and (25) involving the cube of the parallax represent 
the principal part of the tide-producing force. For the moon this is 
about 98 per cent of the whole and for the sun a higher percentage. 
The part of the tide-producing force represented by formulas (24) and 
(26) and involving the fourth power of the parallax is of very little 
practical importance but as a matter of theoretical interest will be 
later given further attention. 
47. An examination of formulas (23) and (25) shows that the prin- 
cipal part of the tide-producing force is symmetrically distributed 
over the earth’s surface with respect to a plane through the center of 
the earth and perpendicular to a line joining the centers of the earth 
and moon. The vertical component (23) has a maximum positive 
value when the zenith distance z=0 or 180° and a maximum negative 
value when z=90°, the maximum negative value being one-half as 
great as the maximum positive value. The vertical component be- 
