602] 
ON THE POTENTIALS OE POLYGONS AND POLYHEDRA. 
269 
and the integral J r 2 dco is therefore 
sin 0 d6 d(f> 
[(cos a cos (f> + cos /3 sin <j>) sin 6 + cos 7 cos 0] 2 ’ 
:os a cos 
and, in particular, if p coincide with the axis of 0, so that the equation of the plane 
is z = p, then the integral is 
9. The integration in regard to 6 can be at once performed; viz. in the latter 
case we have j ~ cog2 q " = sec $ 5 an d i n the former case, writing, as we may do, 
1 fsm(0-N + N) dO 
M 2 ) cos 2 (0 — N) 
I 
[(cos a cos (f> + cos /3 sin </>) sin 6 + cos 7 cos 0] 2 M 2 
a cos <6 + cos /8 sin </>) sin 6 + cos 7 cos 0] 2 M 2 J cos 2 (6 — 
= [cos IV sec (0 — N) + sin iVlog tan (¿77 + \ (0 — iV)}]. 
Case of a Polyhedron or a Polygon. 
10. Consider now the pyramid, vertex the origin 0, standing on a polygonal base. 
Letting fall from the vertex a perpendicular OM on the base of the pyramid, and 
drawing planes through OM and the several vertices of the polygon, we thus divide 
the pyramid into triangular pyramids; viz. AB being any side of the polygon, a com 
ponent pyramid (or tetrahedron) will be OMAB, vertex 0 and base MAB, where MO 
is a perpendicular at M to the triangular base MAB. And drawing through MO a 
plane at right angles to AB, meeting it in D (viz. MD is the perpendicular from M 
on the base AB of the triangle), we divide the triangular pyramid into two pyramids 
OMAD, OMBD, each having for its base a right-angled triangle ; viz. the vertex is 0, 
the base is the triangle ADM (or, as the case may be, BDM) right-angled at D, and 
OM is a perpendicular at the vertex M to the plane of the triangle. It is to be 
observed that, in speaking of the original pyramid as thus divided, we mean that the 
pyramid is the sum of the component pyramids taken each with the proper sign, + or —, 
as the case may be. 
11. In the case of a polyhedron, this is in the like sense divisible into pyramids 
having for the common vertex the origin or point 0, and standing on the several 
faces respectively; hence the polyhedron is ultimately divisible into triangular pyramids 
such as 0ADM, where ADM is a triangle right-angled at D, and where 0M is a 
perpendicular at M to the plane of the triangle. Hence the potential of the polyhedron