•114 MR. AIRY ON THE LAWS OF THE TIDES 



In order to institute a companson of these numerical results with the formula given 

 by theory, it may be convenient to premise the following expressions : — 



If the tide were created in a uniform channel, forming a great circle round the 

 earth, the expression for the height of the water would have the form 



A.sin/?+B.sin2jo+90°, 

 where B would have the same sign as A if the velocity of the tide-wave were greater 



than \/^, h being the depth of the channel. 



If the tide at the mouth of a gulf be a pure tide, or one in which the elevation is 



expressed by a . sin;?, then the elevation at any point in the bay will be expressed by 



the formula 



A . sin ;? +B . sin -Jjo -f 90°, 



where (in the case of a gulf sufficiently long to have a tidal node) A and B have dif- 

 ferent signs from the mouth of the gulf to the node, and afterwards have similar 

 signs. 



If the tide at the mouth of an indefinitely long river bs a pure tide, then the eleva- 

 tion at any point in the river will be expressed by the formula 



A.sinp-1-B.sin2/?, 

 where B has the same sign as A if the section of the river be a parallelogram, but 

 may have the opposite sign if the section expand very much at the top. 



^ „ B . ' n 1 I vertical oscillation of the water 

 In all cases ^ is a quantity of the same order as depth of the water 



I shall now proceed to examine the results deducible from the last Table. 



The stations Kilbaha, Casleh Bay, Old Head, Mullaghmore, may fairly be consi- 

 dered as littoral stations on the open Atlantic ocean. And their formulae agree among 

 themselves almost absolutely to the second term, and in a great measure to the third 

 term. They show clearly that the Atlantic tide there is not a pure tide. But the 

 form of the argument does not agree with either of the two first formulae just cited^ 

 which alone can apply to it. At Castle Townsend, which is nearly as much exposed, 

 but on a different side of the island, the formula agrees pretty well with the first of 

 those above, supposing the depth of the sea very great. 



At Dunmore East the tide has nearly assumed the form of a river tide. 



Proceeding from Kingstown (where the character of the tide is similar to that at 

 Kilbaha, &c.) to Clogher Head, Ardglass, and Donaghadee, the argument of the 

 second term undergoes a progressive change, its phase being less advanced. As the 

 epoch of the first term is absolutely the same at these stations (the tide being simul- 

 taneous at all of them), it appears that the wave represented by the second term is 

 progressive. It seems therefore that it does not originate in the peculiarities of a 

 gulf-tide (contained in the second formula just cited), but that it has been created 

 either on the open sea or in the shallower water between Ireland and Cornwall, and 

 now travels on as an independent wave. 



