HAHMONIC ANALYSIS AND PREDICTION OF TIDES. 51 



= 1/4 6 sm 7 cos^ i/ [_1 + 6 ^^^^ COS 2P + 9 ^^^^^J ^ 



1-3 "^^^ 



cos^ -|-/ 



X COS I d- tan-^ V tan P 



1 + 3^ 



cos^ ii 



= l/2e?^^-^^^5^cos (T+h-s-^U + ^-v + Q) (194) 



in which Qa = same as in (191) 



and 



o cos I 



cos^ hi 2 — 3 tan^ ^7 



Q = tan-^ \ tan P = tan-^ -. — m. — r-fr tan P ci qkv 



jcos_/_ 4 — 3tan2^7 uyo/ 



COS^ t-* 



The values of log Qs, and Q corresponding to different values of P 

 and the mean value of 7 will be found in Tables 9 and 10, respectively. 



In formula (194) the Fof the argument is taken as T+Ti — s — '^j^ and 

 the u SiS^ — v + Q. Formulas (190) and (194) both ref)resent the com- 

 posite Mj tide and are equal to each other, since each is the sum of the 

 terms (187) and (188). It may also be shown from (192) and (195) 

 that 



Qu + Q = P = p-^ (196) 



and therefore , 



p-Qn-^ + Q (197) 



The complete argument of (190) is therefore equivalent to the 

 argument ot (194), the distinction being that the uniformly varying 

 element p in the V of the first argument has been transferred to the u in 

 the latter, where it is assumed to be constant in the analysis of any given 

 series of observations. The speed of the component as determined 

 by the remaining part of the Fis then exactly one-half the speed of the 

 component Mj [term (J.), of (100)]; and with this assumption the 

 summations for component Mj will be adapted to the analysis for the 

 component Mj. It will be noted, however, that the u in this case, 

 unlike the u's of any of the other components discussed, has a pro- 

 gressive forward change that takes it entirely around the circumfer- 

 ence (see Table 10 for values of Q). The true average speed of this 

 component is therefore determined by the Fof the argument of (190),. 

 the approximate average speed determined by the Fof formula (194) 

 being assumed when the summations for component Mj are to be used 

 for Mj. 



In obtaining an expression for the mean value of the variable factors 

 of the coefficient of this component the u from formula (190) must be 

 used. For this coefficient we have 



fsin 7 cos^ il / , /o \"I 



r ■ T ,,;^[C0sQu Sin^u • 11 /ino\ 



= sm 7 cos^ i7 — ^-^ cos v yy^ sinp] (198) 



