ON THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 47 



Now, observation and tteory agree in showing that k" is very nearly 

 •eqnal to K-g ; hence we are justified in substituting ».•, for k" in the small 

 solar declinational term of (8) involving •317H". 



This being so, (8) becomes 



A2=Mcos(24/— /L£) + M,cos(2i//,-/Li,) (10) 



In the equilibrium theory each H is proportional to the corresponding 

 term in the harmonically developed potential. This proportionality holds 

 nearly between tides of nearly the same speed ; hence in the solar tides 

 we may assume (see Sched. B, 1883, and note that cot ^ A^=^cot2 ^w) 

 that, 



cos^ A 1 



___.317H"=3^^ (H._H,)=H„ 



and M^ reduces to 



00^ C COS c 



^'=^SSd, H.+3(P,-1)H = J3^'_H,[1+3(P,-1)] Marly 



=P-'^=. (11) 



Now, since A,=16°-36=16° 22', sec^ A^=l-086, also P,3=p„ and there- 

 fore 



M,=l-086p, cos2 a, H, (12) 



In a similar way, according to the equilibrium theory, we should 

 have 



"3^(Hn— Hi)=H„. 



Although this proportionahty is probably not actually very exact, yet 

 in our supposed ignorance of the lunar elliptic tides we have to assume 

 its truth. Also, we must assume that the two elliptic tides N and L 

 suffer the same retardation, and therefore K-„=ki^£. 



With these assumptions, 



-.-r y-^, - H_ cos k„ — Hi cos »Ci 



H„ + (F-1)^^ ^;^ 'cos(^-0=H^[l+3(F-l)]=H„F3. 



Then, since 



cos'' A^ 

 we have 



,, „ ,„ cos^ a' — cos^A „ 



M=f p' H^+ g-^, ^^ •683H" cos (."-.„), 



cos^c' — cos^ A „^„„ 

 /^=^V+ si^2/^^ •683H"sin(K-"-.„). • ■ • (13) 



If we put 



C,=^ C,= -:S|--x57°-3, 

 2 sm-'A, 2 sm^A^ 



then 



log C, =-6344, log 02=2-3925, 



and Ci, C2 are absolute constants for all times and places. 



