140 TJ. S. COAST AND GEODETIC SURYEY. 



Excepting for the lag, therefore, the flow will be positive whenfthe 

 elevation of the water at M is higher than at N and negative when 

 the water at N is the higher. 



Let 



H, = amplitude of any component A for station M. 

 H„ = amplitude of same component A for station N. 

 K, = epoch of same component A for station M. 

 Ky/ = epoch of same component A for station N. 

 i, = longitude of station M (positive if west, negative if east) . 

 jL,/ = longitude of station N (positive if west, negative if east) . 

 p = subscript of component A. 

 (Fo + i/)/ = local Vo + u for component A at station M. 

 (Fo + 'u),, = local Vo + u for component A at station N. 

 Then (Fo + u)„=(7o + u), + 2> {L,-L,,) (476) 



The equations of the heights of the tide due to component A at 

 stations M and N, respectively, may be written 



2// =fH/ cos [at + (Vo + u) y- K,] 



=fEy cos [at +{Vo + u);\ cos k, + fH, sin [at + ( Fo + u) J sin k, (477) 



2/// =fS,, cos [at +{Vo + u),,- K,;\ 



=fH„ cos [at+{Vo+u)f + 'p {L, - L,^ - k„] 



=fH„ cos [at + ( Fo + ii) ,] cos [k„ -p{L,- X„) ] 



+fH„ sin [at + ( Fo - u) ,] sin [k„ -p{L,- L,,) ] (478) 



The difference in the height due to component A, positive when the 

 tide at M is the higher and negative when the tide at N is the higher, 

 may now be written 



y = y^- 2/// =/ { S, cos K, - H,, cos [k,, -p{L,- L,,) ]] cos [at + {Vo + u) ,] 



+f { H, sin K, - B„ sm [k^, -p{L,- L,) ] } sin [at +{Vo + u),] 



=f^H/ + H,/-2H,H,, cos W-iiu + piLy-L,,)] x 



^^o r^i . ( T/ ^oA i.r.-' g/ sin K, - H,, sin [k,, -p{L,- L,,) ] "] , 



cos at + ( Ko + 'Mj, — tan -^ =b p tr f~Ti (479; 



L ^,cos/f/-^„cos[/c„-2)(i/-i//)]J 



If we let 



E = V^/ + ^// - 25^, H,, cos [k, - K„ + v{L,- i„) ] (480) 



and 



tan-i - ^/ sin «/ - E„ sin [k,, -p{L, -£,,)] ,^^^. 



H, cos K, — H„ cos [k„ — p{L, — L„) ] 



and substitute in (479), we have 



y =fH cos [at+{Vo-u),-K] (482) 



In (481) the quadrant of k is determined by the signs of the numer- 

 ator and denominator, which correspond, respectively, to the signs 

 of the sine and cosine of the angle. 



