Mr. Woolhouse on the Deposit of Submarine Cables. 357 
of two consecutive values k, k+ Ak, and (A,) the mean of the 
second differences which stand respectively opposite to them, 
the increment of z may be calculated from the formula 
ft) 
PN aye d 
a(4)=6)-§ (a9. 
For the computation of * we have 
1 dx_1 ds eaten 
ag er Foose =f cot a; 
but as the value of this quantity is indefinitely great when is 
indefinitely small, the integration by the same method becomes 
impracticable. To obviate this inconvenience, calculate a table 
of the values of 
de) tig?) ea 
aA TM Vhcoto, 
which will not be subject to any abrupt change. Then ~ will be 
P 
the integral of Q aon: P 
Vh 
The value of Q at the lowest point of the curve, where o=0 
and h=0, may be found thus :—Since sy>=1, we shall have in 
the immediate vicinity of this point ¥Y = {dh =h, Also since 
an element of the-curve will coincide with the circle of curvature, 
we shall also have r= “y(2p9—y); or substituting y=pof, 
i= say ak VB when k=0. 
Now the quantities Q being differenced, and A,, A,, &c. de- 
noting the differences which immediately succeed a given value 
of h, and representing the values of A by ordinal numbers q, so 
that q= oh, 
AW 
becomes g +7, will be Q+7A,+ 
the value of Q when h becomes h+iAh, or when g 
nee A,+ &c.; and the in- 
crement of — in passing from f to h+Ah, or from g to q+], 
ry oe BY 
A—= ————— +7A + A,, &e. . 
me W/W Ah L thaens Q 1 5) 2 
* For the integration of the terms of this expression I have arrived at 
the following curious general form :— 
eet, 2.4...2n gle tice i\-2, ta te be 
ar = 2.55 maple {q (145) expandedasfaras t } 
Phil. Mag. S. 4. Vol. 19, No, 128. May 1860. 25 
