HARMONIC AISTALYSIS AND PREDICTION OF TIDES. 53 



of a factor error which has been applied alike to all amplitudes for 

 this component. Neither has the use of these components in the 

 prediction of the tides led to any error in that work, since the error in 

 the factor for reduction to the mean value has been exactly compen- 

 sated by a corresponding error in the reciprocal factor used for reduc- 

 ing the mean value to the amplitude for the year of prediction. 

 Therefore no practical difficulties have resulted from this error. To 

 change now to the corrected formula, unless the change were uni- 

 versally adopted, would lead to considerable confusion in the com- 

 parison of the amplitudes as determined and published by different 

 authorities. It seems wisest, therefore, for the time being to adhere 

 to the present practice of using formula (204) or its approximate 

 equivalent 



sin 0) COS^ 1 03 cos* ^i ^ cn^ c rw . /\ r:^c^n\ 



sin/cosH/ xg.-(FofO.)xa. (206) 



for the reduction of the component M^. 



The resulting amplitudes may at any time be readily converted into 

 the corrected means by the application of the factor 1.42 from (205). 



15. TIDES DEPENDING UPON THE FOURTH POWER OF MOON's PARALLAX. 



A reference to equations (74) and (75) , pages 28-29, shows that the 

 tide depending upon the fourth power of the moon's parallax is very 

 small, the maximum value being only about 2 per cent of the total 

 lunar tide. In developing the term representing this tide we need, 

 therefore, seek only a rough approximation to its true value, neglect- 

 ing the elements which are relatively small compared with the entire 

 term. As the angle / is never very large, the sine will always be 

 smaller than the cosine, and for our approximation the powers of sin 

 / and sin \ I above the first may be neglected in this development. 



Substituting the value of cos B from (81) in the second term of (74) 

 and neglecting powers of sin / and sin \ I above the first, we obtaiD 

 the following : 



3/2 ^ J, [5/3 cos^ - cos 01 



= 3/2 -p -5-4 [5/3 {3 sin X cos^ X sin / cos* ^ I sin I cos^ {l — x) 



+ cos^ X cos^ ^ I cos^ (Z — x) } — sin X sin / sin I 



— cos X cos^ i I cos (l — x)] 



= 3/2 -w ji [5 sin X cos^ X sin I cos* ^ /{^ sin Z— 1/4 sin (Z — 2%) 



+ l/4sin(3Z-2x)} + 5/3cos3XcosH^{3/4cos(Z-x) 

 + 1/4 cos 3 (Z — x) } — sin X sin / sin Z 



— cos X cos^ ^ I cos (Z — x)] 



= 3/2 ^ ^, [5/12 cos^ X cos« ^ / cos 3 (Z-x) 



+ 5/4 sin X cos^ X sin I cos* ^ I cos (3Z — 2x — tz/^) 

 + 5/4 sin X cos^ X sin / cos* -J / cos (Z — 2x + tt/j) 

 + {5/4 cos^ X cos^ I /—cos X cos^ ^ /} cos (Z — x) 

 + {5/2 sin X cos^ X sin 7 cos* J / 

 -sin X sin /} cos {1-t/^)] (207) 



