TIDAL OBSERVATIONS. 385 



"We have 



*=c'cos2^cos(2xNP')=ccos«5(2cosi!NP'-l). 



But 



cos« S cosS NP' = cos-! SP' =. (cos VS cos TV + sin I'.S sin VP' cos t')-' 



= { htpi C03 (VP'-YS)+ l^P CO.S (VP'+rS) } ' 

 = (l±^)%os3 (rP'-YS)+'^\co32 YF- 8mn^S)+C=|^y cosHYP'+YS) 

 = j{ (l+p^)%os2(YP'-YS)+^-^'cos 2YF+(^^y oos2(YF+YS) - sin^^J 



= J I (b:|^^)%os2(YP'-YS)+^-^'co92YP' + C-f 'O'^'o^^a^^ 



Hence 



,=., I (y:|^)%os2(YP'-YS)+^-^cos2YF+(i:^y cos2(YF+YS)} . (l) 



If time be reckoned from the transit of the first point of Aries across the 

 meridian of P', we have 



YP'=y« 



when the formula is applied to the solar tide, and for the lunar 



YP' = y<-n, 



where tl denotes the right ascension of the intersection of the moon's orbit 

 with the earth's equator, from the first point of Aries. For the solar tide 

 YS is the sun's longitude, and for the lunar YS is equal to the moon's longi- 

 tude with a correction depending on LI. Hence, in the two cases respectively, 

 we have ^ ^ 



YS = JJ!' + 6 + P, YS = (Tl' + €'+Q (2) 



where e, e' denote the longitudes of the two bodies at the time t=0, P the 

 sun's elliptic inequality of longitude, and Q, the moon's elliptic and incli- 

 national inequality of longitude. For the mean semidiurnal and the decli- 

 national semidiurnal tides we neglect these inequalities, and so find 



(Solar) "j 



s = c { (L+yi^ycos2[(y-„y-e]+?^ cos 2y#+ (i=fi^)%o3 2[(y + „V+s] } , 



where u denotes the obliquity of the ecliptic, and 



(Lunar) 



,. .e' { (i+|5!iycos 2[(y-<T)^-e']+*4^' COS 2yt+(^-^)%o, o.[iy+<f)t+e'] } • ^ 



Denoting by E, S, 31 the masses of the earth, the sun, and the moon, by 

 «■, -or' the parallaxes of the sun and moon, expressed in radial measure 



D'-«-rS]' ^y « t^® ®^^^^'^ ^^<^""' ^^^ ^^ ^ ^^^^ latitude of the place, 

 1872.'"' 2 D 



(3) 



