4 REPORT—1849. 
The cylinder was deflected over to various angles by means of a weight attached by 
a string to the arm of a lever fixed to the cylindrical model. 

Angle of deflection. Angle to which the model vibrated. 
fe} ‘ ie} ‘ 
22 °30 ‘ 22 24 
2210 22 6 
2) .b4 21 48 
21 36 21 Va 
&e. &e. 
When the cylinder oscillated, in all circumstances the same as above, except 
being surrounded by salt water, the amplitude of oscillation was as follows :— ; 



























Angle of deflection. Angle to which the model vibrated. 
22 30 22 0 
21 36 21 3 
20 48 20 16 
&e. &e. 
Clearly showing that the amplitude of vibration, when oscillating in water, is con- 
siderably less than when oscillating without water: in the above instance there is a 
falling off in the angle of amplitude of 24’, or nearly half of adegree. This amount 
has been confirmed by several experiments made with great care; and it appears 
only fair to attribute this decrease in the amplitude of oscillation to the circumstance 
of the friction of the water on the surface of the cylinder. The amount of force 
acting on the surface of the cylinder necessary to cause the decrease in the ampli- 
tude of oscillation shown by the experiment was calculated, and the author thinks 
that this amount of force is not equally distributed on the surface of the cylinder : 
in consequence of this he thought the amount on any particular part might vary as 
the depth. Onthis supposition a constant pressure at a unit of depth is assumed ; 
this, multiplied by the depth of any other point of the cylinder immersed in the 
water, will give the pressure at that point. These forces or moments being summed 
by integration and equated with the sum of the moments given by the experiments, 
we shall have the following value of the constant pressure at a unit of depth, 
*0000469. This constant in another experiment, the weight of the model being 
197 lbs. avoirdupois, (and consequently the part immersed in the wafer was very 
different from the other experiment) was ‘0000452, which differs very little from the 
former; showing that the hypothesis assumed in computation is not far from the 
truth. 
On Elliptic Integration. By Ropert Rawson.~ 
The object of this communication is, in the first place, to change the form of the 
elliptic function from that involving the square of the sine of the amplitude to a 
form involving simply the cosine of the amplitude, by means of the well-known — 
trigonometrical formula, that twice the sine of half an are squared is equal to unity 
minus the cosine of the same arc. ; 
The author believes this form of the elliptic function to possess several advantages, — 
and therefore would be more useful to tabulate than the form of Legendre, whose — 
tables are not in a good practical form for use. With a view to tabulate this func- — 
tion in a more extensive manner than Legendre has done, several investigations — 
have been_made to compare the functions of the first order with different amplitudes 
and moduli; a formula for this purpose has been obtained where the relation be- — 
tween the amplitudes is much more simple than that discovered by Lagrange. 
A different mode of investigation has been pursued by the author than that pur- 
sued by Lagrange, Legendre, Abel, or any authors who have written on this sub- 
ject; and by taking a relation between’ the amplitudes expressed by means of an 
unknown function of one of the amplitudes, we are conducted to tw6 equations, the 
first of which is an elliptic function of the first order, equal to a constant times an- 
other integral of an arbitrary character, and the second a functional equation, which 
must be satisfied, between the function assumed in the relation of the amplitudes and 
i 
J. 
