12 U. S. COAST AND GEODETIC SURVEY 
Attraction of moon for unit mass at point P in direction po=e (3) 
37. Let each of these forces be resolved into a vertical component 
along the radius OP and a horizontal component perpendicular to the 
same in the plane OPC, and consider the direction from O toward P 
as positive for the vertical component and the direction corresponding 
to the azimuth of the moon as positive for the horizontal component. 
We then have from (2) and (8) 
Attraction at O in direction O to p= COS 2 (4) 
Attraction at O perpendicular to op=" sin 2 (5) 
Attraction at P in direction O to pate cos CPR (6) 
Attraction at P perpendicular to op="= sin CPR (7) 
38. The tide-producing force of the moon at any point P is measured 
by the difference between the attraction at P and at the center of 
the earth. Letting 
F,=vertical component of tide-producing force, and 
F,=horizontal component in azimuth of moon, 
and taking the differences between (6) and (4) and between (7) and 
(5), we obtain the following expressions for these component forces 
in terms of the unit p: 
F, ju= (°° a *) (3) 
Pe [n= M(t") (9) 
39. From the plane triangle COP the following relations may be 
obtained: 
b=r?+ @2—2rd cos z=d*[1—2(r/d) cos z+ (r/d)?| (10) 
: ; sin z 
sin CPR=sin CPO= (d/b) sin 2=—9G/d) eee (11) 
bad coe cos z—r/d 
cos CPR= (1—sin OGD i= =r mae GED. (12) 
40. In figure 2 it will be noted that the value of z, being reckoned 
in any plane from the line OC, may vary from zero to 180°, and also 
that the angle CPR increases as z increases within the same limits. 
Sin z and sin CPR will therefore always be positive. As the angle 
OCP is always very small, the angle CPF will differ by only a very 
small amount from the angle z and wil] usually be in the same quad- 
rant. In obtaining the square root for the numerator of (12) it was 
therefore necessary to use only that sign which would preserve this 
