224 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



impossible to draw the isotherms, even those strictly 

 relating to Liverpool Bay. 



Interpolation. It is quite necessary to interpolate 

 values for the surface temperatures, when drawing the 

 isotherms over a large area, since it is not generally 

 practicable to make the observations during a period of 

 time short enough to preclude the possibility of a change 

 of temperature. Even in the "Western side of the Irish 

 Sea it has not been possible to visit all the stations 

 considered during the same cruise ; and it was not always 

 practicable to arrange that the cruises of the " Helga " 

 and the "James Fletcher" should be made during the 

 same days. Since the stations visited by the former vessel 

 are much more numerous, and cover a larger area than 

 the Lancashire ones, they have been taken as the basis 

 for the construction of the isotherms, and the observations 

 made on the English side of the Channel have been 

 reduced so that they represent the temperatures which 

 were, probably, characteristic of the sea during those 

 days on which the " Helga " made her observations. Just 

 how best to reduce these observations was a matter of 

 some little difficulty. I tried at first to make use of the 

 now well-known formula for interpolation suggested by 

 D'Arcy Thompson,* that is 

 fit) = A + A t sin (6 + e x ) + A 2 (sin 2 + e 2 ) + &c. 

 In this expression the time is regarded as an angle, 

 the year being supposed to consist of 360 days, and the 

 temperature at any date is then a function of the sine of 

 the angle. Now if we had only four observations for the 

 whole year, these observations being made at the times 

 of the maximum, minimum, and half-amplitude, this 

 formula would afford the only possible means of inter- 



* Report (Northern Area) on Fishery and Hydrographical Conditions 

 in the North Sea and adjacent waters. (1904-1905); Cd. [3358], 

 p. 18G, 1907, 



