68 



28 ; transforming the functions of /, and the expressions of u above 

 listed into functions of A^; and finding the mean value of these 

 expressions as N varies from to 360°. This somewhat lengthy- 

 derivation, which need not be here repeated, shows that the prod- 

 ucts of these mean values are as follows: 



The numerical values in the last column are found by substituting 

 the values of a)=23°.452; and 'i=5°.145 in the expressions in the 

 second column. It may be noted that as the moon's orbit tilts to 

 and fro, the median value of its inclination I to the Equator is the 

 inclination co of the ecliptic to the Equator. Since the values of 

 cos* Yii and of (1 —3/2 sin- i) are very close to unity, the mean values 

 differ but little from the value of the function when /=co. 



The expressions for the reduction factor F are then : 



For Ma, Na, 2N, V2, X2, and ^2, 

 For Oi, Qi, 2Q, and pi, 

 For 00, 

 For Ji, 

 For Mf, 

 For Mm, 



F=0.9154/cos* %I 

 F=0.3800/sin/cos2 YJ 

 J^=0.0164/sin/sin2 yj 

 i^=0.7214/sin 2/ 

 jP=0.1578/sin2 / 

 2^=0.7532/(1-3/2 sin^ I) 



(94) 



126. The reduction factors for the lunisolar components K] and K2 

 are more lengthy functions of /, and those for the L2 and Mi compo- 

 nents are still more complicated. The derivation of these factors is 

 explained in full in Special Publication No. 98; United States Coast 

 and Geodetic Survey, and is not here described. 



127. Application of reduction factors. — The logarithms of the reduc- 

 tion factors for the several lunar and lunisolar components, corre- 

 sponding to each tenth of a degree of /, are tabulated in Manuals on 

 the Harmonic Analysis of Tides, special tables being included from 

 which the factors for the L2 and Mi may be found. To find the mean 

 value of the amplitude of a component from the value determined 

 from a particular set of observations, the value of N at the middle of 

 the period is taken off as described in paragraph 118, the corresponding 

 value of / taken from a table, and from it the logarithm of F ascer- 

 tained. Thus the amphtude of the M2 component at Sitka, Alaska, 

 for the 29-day period beginning July 1, 1893, corrected by the aug- 

 menting factor, was found in paragraph 99 to be, i?=3.430 feet. 



