400 
Proceedings of the Eoyal Society 
It may here be asked, Is this lifting power of the surface currents 
sufficient to account for the vertical circulation which we find in 
the ocean ? In all probability it is not. There seems to be no 
reason why this vertically rising current under the equator should 
draw its supplies from the furthest limits of the ocean, which it 
would require to do to explain the conditions we find existing. 
Yet there can be no doubt but that these horizontal surface cur- 
rents really do assist in producing a vertical circulation. 
3. On a New Investigation of the Series for the Sine and 
Cosine of an Arc. By Edward Sang. 
The sines of the successive equidifferent arcs form a progression 
having for its general character the relation 
<f>n-l — 2 <f)7l + (f>n+ 1 — 4>n • V , 
and the properties of sines may be deduced from this general for- 
mula. Viewed in this light, the angular functions become cases 
only of more general ones. 
If we suppose A, B, C to be three consecutive terms of such a 
progression we must have 
A - 2B + C = vB, 
from which, when three of the four quantities, A, B, C, v are given, 
the fourth may be found. Let then A and B, the first and second 
terms of the progression, and v the common coefficient, be given ; 
the succeeding terms may be computed thus : — 
<f> 0 = A 
- A +B 
+ Bu 
<f> 1 = 
B 
-A 
+ B(l+v) 
— Av 
+ B( +2v + v 2 ) 
rf> 2 = — A 
+ B(2 + v) 
-A(l+v) 
+ B(1 + 3u + u 2 ) 
— A( + 2v + v 2 ) 
+ B( + 3v + 4u 2 -l- v s ) 
