as are terminated by Planes , &c. 277 
begin at that value of x, no correction is to be added. The 
last of the arcs, in the expression above, is the complement 
of the first ; denote then the first by A, and we have 
A = 4 (x -f a) A — 2 xtt -f- 4, V~ V 2k -f « arc (sine = ; 
lastly, if we want the whole fluent, when x = 2k , we get 
A = 27 T (v/ zka. -j- a* — a). 
Cor. 1. If we make u, infinite, in this last value of A, it be- 
comes A = 2&7T ; which is the action of an infinitely long cir- 
cular cylinder , on a point at its surface. This is the attraction 
of the whole cylinder when x — 2k\ to find the same for any 
value of x, make a. infinite in formula (/ 3 ) ; this gives 
A — 4.2: arc (tang. = j + 4 arc (sine = 
— 4<* arc {sine = (1 + ^7=$ 5 but (Euleri Calc. DifF. 
P- S 7 6 ) 
Arc {sine = (1 -f- -) — arc (sine = + 
C. 2a j y' 2 £ ) v. V zk ] 
zkx- 
20 , 
qu. prox. ; the substitution of this value gives 
A = 4-r arc (tang. = 
v' zk—x 
V x 
-f> 4& arc (sine = 2s/ 2kx-x % i 
which, because the latter arc is the complement of the former, 
is changed to 
A = 2&7T- — 4 ( k — x) arc (tang. = 
dzk-. 
\J~X 
3 ✓ %kx — X\ 
Cor. 2. In like manner we may find the attraction of an in- 
finitely long parabolic cylinder , on a point in its surface, at the 
vertex of the parabola ; this is effected by making k infinite in 
formula (/ 3 ), whence there results 
A = 4# arc (tang. = —-42 arc (tang. = — -{- 4 V ax; or 
mdcccxii. O o 
