II. LENGTH. 71 



trolling the other) moving in accordance with the level of the water on a disc 

 of glass (M). This disc is turned round its axis by the clockwork in 6 hours. 

 The axis of the rollers J, J is parallel to the direction of their motion on the 

 disc M. The number of the revolutions of the rollers is noted in certain 

 periods by means of the divided rim (the rim is divided in 100 parts, tenths 

 being estimated) and a numerical apparatus showing revolutions up to 100. 



The height of the water may be taken from the point when the roller J 

 stands in the centre of the disc M. 



If the height of the water taken from this point be x, and the diminution of 



the movement of the buoy by the pinion F be - 



.... (I.) 



Is the expression for the movement of a point in the circumference of the 

 roller J in a period during which the disc M revolves through an arc </>. 



The expression jx8<f> is the area of a figure with the ordinates x and the 

 base </> (x representing the height of the water and <f> the time). The mean 

 value of .r or the height of a rectangle of equal area with this figure and the 

 base </> is found by dividing by (j>. This mean value of x is the mean height 

 to be found, equal m suppose. Then 



J^_ (2 .) 



The movement of a point of the circumference of the roller J is equal to 

 -JxSQ. This is equal to the product of the circumference of the roller J 



called p and the difference of readings on the margin of the roller J at the 

 beginning and end of the period. Hence, if the readings are called a l and Oj, 



.-I**-'* .... (3.) 



If z is the number of seconds corresponding to the arc </>, and b the con- 

 stant arc through which the disc M revolves in a second: 



(4.) 



Finally, if ~ = 



the mean height of the sea. 



The single constant c is easily to be determined by experiment and also 

 with great exactness without the knowledge of the dimensions of the appa- 

 ratus, as follows : In a certain position of the roller J a number of revolutions 

 is made by the disc M, representing a number of seconds z l (a revolution is 

 made in 21,600 seconds). At the beginning and end of these revolutions the 

 readings (a x and a 2 ) of the roller J are noted. After this a certain length (/) 

 of the copper wire upon the disc C is unrolled (measured by means of the 

 circumference of the disc C, equal 2 meters, and the index, or directly). In 

 the new position of the roller J a number of revolutions of the disc M again 

 is made, representing a certain number of seconds (z 2 ), and also the corre- 

 sponding two readings (a 3 and cr 4 ) of the position of the roller at the beginning 

 and end of these revolutions are now observed. All the requisite data for the 

 determination of c are now obtained. If the two values of m corresponding 



