50 - MH. AIRY ON THE LAWS OF THE TIDES 



and the question now is, how this function can be represented, through the course of 

 one tide, by a formula similar to 



Ao+Ai sin phase +A2 sin 2 phase + &c. 

 +Bi cos phase +B2 cos 2 phase + &c. 

 For this purpose I have taken the mean value of the function for each sixteenth part, 

 and for each twelfth part, of the entire circle of phase, and have combined these num- 

 bers according to the rules just laid down. The result is that 



32 16 



The sum —A. is increased by— X 0-3980 X«. 



"' O 1 ^ 



The sum — B. is increased by - — X 0-0392 X h. 



32 16 



The sum — A, is increased bv — X 0-8284 X a. 



32 16 

 The sum — B., is increased by X0-1648X^. 



IS ^ ''IS 



32 32 16 



The sum — A, +t- Ao is increased by — X 2«. 



TS '■ OTS ^ "IS 



32 32 16 



The sum — B, - ^ B. is increased by — — X 0*8284 X b, 



24 12 



The sum — Ao is increased by —X 0-5360 X<i. 



IS ^ "^ IS 



24 1'^ 



The sum — B3 is increased by — ^ X 00704 X h. 



2 

 The mean, or Aq, is increased by -f - 6. 



The corrections are to be applied with opposite signs, in order to free from the effects 

 of diurnal tide the results given by the observations. 



Another cause for which a correction is due is, the difference of height at the be- 

 ginning and at the end (the correction for diurnal tide being previously applied). If 

 the whole rise c be supposed to have come by uniform degrees, the effects produced 



32 32 32 32 24 



in the sums -^ ^\-, v^2> "^1 + 3^ A3, and —A3, are — c, —2c, —4c, and — c. The 



corrections must have opposite signs. 



In this manner (confining ourselves for a moment to the consideration of 4 phase) 



32 32 



we have such expressions as — A4 and — B4. And the quantity which we wish to ob- 



tain is A4 sin 4 phase 4-B4Cos 4 phase, which may be converted into one of this form, 



T //32 . \^ , /32„ \^ 

 = 32V(tA4)+(vB4)- 



T» 



^ K^-\-^^y, sin . 4 phase ■\-<p^ where tan ^= -r- The coefficient is 



