18 U. S. COAST AND GEODETIC SURVEY. 



Let each of these forces be resolved in directions parallel and per- 

 pendicular to the radius through P, and let the direction from 

 toward P be taken as positive and the reverse as negative, and also 

 the direction of the perpendicular to OP that most nearly conforms 

 with the direction from toward C as positive, and the reverse 

 direction as negative. We then have from (6) and (7). 



Attraction at in direction to P = -n^ cos (8) 



Attraction at perpendicular to 0P = -^ sin 6 (9) 



Attraction at P in direction to P = -p- cos CPU (10) 



Attraction at P perpendicular to OP = -jrr sin CPR (11) 



As the tide-producing force of the moon at the point P is measured 

 by the difference between the attraction of the moon at P and at 0, 

 the following may be obtained from (8), (9), (10), and (11). 



Tide-producing force at P in direction to P 



= hM[^, cos CPR- J, cos d] (12) 



Tide-producing force at P perpendicular to OP 



= MJf fp sin CPR- J, sin ^1 (13) 



By a solution of the plane triangle COP the following relations are 

 obtained : 



d^ 



h^ = r^ + d^-2rd cos ^ = (^2 h _2 ^ cos e + L^ \ (14) 



• m^jy • nr^n ^ ■ - sin 



sm CPR = sm CPO = t sm 



h r, ^r , , rHi (15) 



[l-2jcos. + j:]* 



r 



cos & — -J 



cos CPR =^1- sin' CPR 



[l-2^cos. + g* 



(16) 



In Figure 7 it will be noted that the value of d, being reckoned 

 from the line OC in any plane may vary from zero to 180°, and also 

 that the angle CPR increases as 6 increases within the same limits. 

 Sin d and sin CPR will therefore always be positive. As the angle 

 OCP is always very small, the angle CPR will differ by only a very 

 small amount from the angle d and will usually be in the same quad- 

 rant. In obtaining the square root for the numerator of (16) it was 

 therefore necessary to use only that sign which would preserve this 



