r 35 3 
PROPOSITION V. 
Fig. 5. If the opake prolate fpheroid BPOD, 
given in fpecies and pofition, be qppofed to the 
given luminous fphere HK QI at the given diflance 
C S, forming the fhadow Ffbc : It is propofed to de- 
termine the figure of the fed ion a R N made by a 
plane, cutting the fhadow perpendicularly to its 
axis at the given diftance M S. 
Let the required curve ^RN be conceived to be 
generated by the extremity R, of the variable right 
line MR, revolving about the given point M as a 
center, the line M R being always perpendicular to 
the axis of the fhadow MS : Let the right line RQ be 
a tangent to the fphere HKQI in the point Q, and in 
the fame plane with the right lines R M, M S, it 
will then reprefen t one of the rays of light, which 
conflitute the conical fuperficies of the fhadow, and, 
therefore, by the laws of optics, will be a tangent to 
the fpheroid alfo ; now when the generating point R 
has arrived at N, the ray RQ^_(being fuppofed to 
revolve with it) will co-incide with the tangent NK, 
touching the fphere in K, and the fpheroid in F : Join 
K, S, and the angles N M S, and N K S, will be 
right angles ; let the fpheroid be fuppofed to be cut, 
by the quadrangular plane N M S K, forming thereby 
the elliptic fedtion BOD, draw C k perpendicular, 
and Cl parallel to NK; put CA = t, M C = ^ 
CS = d, MS = A, SK = r, MN = v, CO = «, 
r 
the fine of ^NM = V, its cofine = V, the fine 
of A C B = Z, its cofine = Z, and radius =2 1 : 
F 2 Then 
