HAKMONIC ANALYSIS AND PREDICTIOISr OF TIDES. 117 



Q 



Let - = the acceleration in the high waters of component Mj due 

 to the presence of component M4. With the M4 wave symmetrical! v 

 situated with respect to the Mg wave, - will also equal the retarda- 

 tion in the low water of component Mj, due to the presence of com- 



a 



ponent M4, and — will equal the total amount by which the duration 



of rise of the tide has been diminished by M4. If the duration of 

 rise has been increased, 6 will be negative. 

 Then, for a maximum of (449) 



a 



and this value of t must satisfy equation (451). 

 Substituting in (451), we have 



aMs sin (2mr-d) +2aM^ sin (4:mr - 2a + ^ - 29) = 

 -aMa sin d-2aM^ sin {2d + 2a-^)=0 



(454) 



But 



2a: - 13 - - 2M2° + M,° (455) 



From (446), 



-2M2°+M4°=±90° 



180° 

 according to whether the duration of rise is greater or less than - 



or whether 9 is negative or positive. 



Then 



2a-8=T90° (456) 



according to whether 9 is positive or negative. 

 Substituting this in (454) 



- aMa sin 9 ± 2aM^ cos 29 = (457) 

 and 



^4 , sin g . 



M'"^' cos 29 ^^''^^ 



the upper or lower sign being used according to whether 9 is positive 

 or negative. As under the assumed conditions 9 must come within 



M 



■the limits ±45°, the ratio of y^ as derived from (458) \\'ill always be 



positive. 



180° 

 The duration of rise of tide due solely to the component M2 is 



a 



The duration of rise as modified by the presence of the assumed M^ 

 :is 



DR = ^^-^ (459) 



a a 



Therefore 



9 = ia80°-aDR) (460) 



