VOL. tvn.] ' PHILOSOPHICAL TRANSACTIOKS. 376 



From c, the centre of the ellipsis, let ce be drawn parallel to the tangent tg, 

 meeting the ellipsis in the point e; and cg perpendicular to the line bcd, meet- 

 ing the tangent in the point g : put the sine of acb = z, its cosine = z', the 

 sine of tgc = v, its cosine = v', radius being unity, ac = <, co = x, and i" 

 — x' = 9"; then will the sine of oce (= the sine of ocd + dce) be expressed 



tx. L 1 • 



by z'v + zv'; and by the last prop, ce = ,-■ -— ^, , ; — ~,i but, by comes, 



CB X c>i = CO X CA, whence we obtain ck ■=■ "^ i^ — (t?^ Y. (z'v + zv')^. 



Prop. 4. Fig. 12. In the two similar right angled plane triangles hks, hmn, 

 right angled at k and m, there is given the right lines ks and ms, to find the 

 acute angles, supposing the given angle hm& to be nearly equal to the required 

 angle hnm. 



Put MS = A, KS = r, MN = u, the sine of the given angle Atom = q, its co- 

 sine = q, the sine of hnm =. v, its cosine = v', the sine of (hnm — him) = x, 

 and radius = 1. Let ml be drawn parallel to hk, and m/ parallel to sk: then 

 in the right angled plane triangles nm/, sml, we have as rad. (l) : mn (u) :: sine 

 HNM (v) : m/ («v), and as rad. (l) : ms (A) : sin. lms (v') : ls (Av'); but ttl 

 + LS + = KS ; therefore uv + Av' = r, and by the foregoing rotation v = a 

 •+- (^x, and v' = q — qx\ therefore these values of v and v' being written in the 

 above equation, we shall find x = 



^ — . ^"T ' ; hence v = 7^^'^ » and v' = T ~ "^ • 



ql\ — qxj (j a — qu ' q A — n » 



Prop. 5. Fig. 13. If the opaque prolate spheroid bpod, given in species and 

 position, be opposed to the given luminous sphere hkqi, at the given distance 

 cs, forming the shadow p/bc: it is proposed to determine the figure of the sec- 

 tion aRN made by a plane, cutting the shadow perpendicularly to its axis at the 

 given distance ms. 



Let the required curve grn be conceived to be generated by the extremity r, 

 of the variable right line mr, revolving about the given point m as a centre, the 

 line MR being always perpendicular to the axis of the shadow ms : let the right 

 line RQ be a tangent to the sphere hkqi in the point q, and in the same plane 

 with the right lines rm, ms ; it will then represent one of the rays of light, which 

 constitute the conical superficies of the shadow, and, therefore, by the laws of 

 optics, will be a tangent to the spheroid also; now when the generating 

 point R has arrived at n, the ray Ra (being supposed to revolve with it) will co- 

 incide with the tangent nk, touching the sphere in k, and the spheroid in f: 

 join K, s, and the angles nms, and nks, will be right angles; let the spheroid be 

 supposed to be cut, by the quadrangular plane nmsk, forming the elliptic section 

 BOD; draw ck perpendicular, and c/ parallel to nk; put ca = /, mc = i, cs = 

 d, MS = A, SK = r, MN = u, CO = X, the sine of Zjnm = v, its.cosine = v' the 

 sine of acb = z, its cosine = z', and radius = 1 : then in the right angled 



