32 



little during one revolution of the component radii, the curve repre- 

 senting the resultant of the two components evidently takes the general 

 form of a sinusoidal curve, with an amplitude slowly fluctuating be- 

 tween the sum and difference of the amplitudes of the components, 

 as shown in figure 21. 



Figure 21.— Tide curve of two components whose speeds are nearly equal. 



If Ay is the amplitude of the major component and b the numerical 

 difference between the speeds of the two components, the equation of 

 the resultant has the form: 



y^Ai cos (ait-\-ai)-\-A2 cos [(ai + ^)^+«2] (33) 



when the speed of the minor component is the greater, and : 



y=Ai cos {ait-i-ai)-\-A2 cos [((21 — 6)^+0:2] (34) 



when the speed of the minor component is less than that of the major. 

 From the discussion in paragraph 55 it is apparent that in the first 

 case, (equation 33), the speed of the resultant is greater than that of 

 the major component when the amplitude of the resultant is large, 

 and less when it is small. The high and low waters of the resultant 

 shown in figure 21 will then progressively lead those of the major com- 

 ponent when the amplitude of the resultant is large, and progressively 

 drop back again when its amplitude is small. In the second case 

 (equation 34), the high and low waters of the resultant will progres- 

 sively lag behind those of the major component when the amplitude 

 of the resultant is large, and progressively catch up with them when 

 the amplitude is small. By taking the sum of the three components: 



y=AiCOS {ait-^a])+A2C0s [(ai + 6)^ + 0:2] +^3 cos [(«! — 6)^ + 0:3] (35) 



the timing of the high and low waters of the resultant may be made, 

 by the selection of the relative values of A2 and A3, to conform to a 

 systematic variation from the timing of the high and low waters of the 

 principal component. If A2 is equal to A3, the timing of the high 

 and low waters of the resultant will conform exactly to those of the 

 simple harmonic component Ai cos (ait-{-a), since the speed of the 

 resultant of the last two terms in equation (35) is, as shown in para- 

 graph 56: 



}U(«i+&) + («i-&)]=«i. (36) 



