HABMONIC ANALYSIS AND PREDICTION OF TIDES. 17 



versely as the square of the distance, it is necessary, in order to rep- 

 resent the attraction at on the same scale, to take a line CQ of 

 ;such a length that GQ:GF = 'CP^:C&- 



The line PQ, joining P and Q, will then represent the direction 

 and magnitude of the resultant force that tends to disturb the posi- 

 tion of P relative to 0, for it represents the difference between the 

 force PC and a force through P equal and parallel to the force Q,C 

 which acts upon 0. This last statement maj^^ be a little clearer to 

 the reader if he will consider the force PC as being resolved into a 

 force FT) equal and parallel to Q,C, and the force FQ. The force 

 FB, acting upon the particle at P, being equal and parallel to the 

 force QC, acting upon a particle at 0, will have no tendency to change 

 the position of P relative to 0. The remaining force FQ mil tend 

 to alter the position of P relative to and is the tide-producing 

 force of the moon at P. The force FQ may be resolved into a vertical 

 •component PP, which tends to raise the water at P, and the hori- 

 _zontal component FT, which tends to move the water horizontally. 



a* 



Fig. 7. 



If the point F' is taken so that the distance CF' is greater than 

 the distance CO, the tide-producing force F'Q' will be directed away 

 from the moon. While at first sight this may appear paradoxical, 

 it will be noted that the moon tends to separate from P', but as 

 is taken as the point of reference, this resulting force that tends to 

 separate the points is considered as being applied at the point P' 

 only. 



We will now seek analytical expressions to represent the tide-pro- 

 ducing force of the moon at any point P within or on the surface of 

 the earth. Referring to Figure 7, 



let r = OF = distance of P from center of earth, 

 h=FC = distance of P from center of moon, 

 d = OC = distance from center of earth to center of moon, 

 d== {7(9P = angle at center of earth betv/een OF and OC, 

 Jf =mass of moon, 



n = attraction of gravitation between unit masses 

 at unit distance apart. 



Since the force of gravitation varies directly as the mass and in- 

 versely as the square of the distance, 



Attraction of moon for unit mass at point in direction C= —4- (6) 

 Attraction of moon for unit mass at point P in direction FC=-^ (7) 



