376 PHILOSOPHICAL TRANSACTIONS. [aNNO I767. 



plane triangles c/s, it will be as rad. (1) : cs (d) :: sine sc/ {V) : si {d\'), and 

 consequently cA (=KS — si) = r — dv'; butby prop. 3,cA= V o — *>» x {t'v + zv')S 

 hence r — dV = v'^^ — <?>' x (z'v + zv')' : now by prop. 1, we find ^ = i' X (/> 

 + 6'z), z = -r== ——- and z' = -7— :. . ; by prop. 2, t^ =, 



— ■^, and ^ = —^—ff~ ; lastly by prop. 4, v = -^^7:, andv = j^-3^^; 

 which values, being substituted in the above equation, will exhibit the nature 

 of the required curve grn, in terms of z and v. 



Scholium. If the sphere hkqi represent the sun, and the spheroid epod one 

 of the primary planets, it will appear, from the preceding reasoning, that the 

 figure of the section of its shadow received on a plane, which is perpendicular to 

 its axis, will not be a circle, except when the axis of the planet produced passes 

 through the sun's centre, but a curve of the oval kind, whose species will be 

 known from the foregoing equation. If the sphere hkqi had been regarded as 

 a spheroid in the above solution, it is easy to see that the foregoing process 

 would have determined the nature of the required curve; but the figure of the 

 sun is so nearly spherical, that it was not thought necessary to embarrass the so- 

 lution with that consideration. 



Hence the duration of an eclipse of a given satelles may be determined in the 

 following manner: let brc (fig. 14) be the section of the shadow, through which 

 the satelles passes, n/jn the path of the satelles, making the given angle n/jm, 

 with the circle of latitude r/)m; bmc a part of the primary's orbit produced, 

 and up the given latitude of the satelles at the time of the syzygia; the circle of 

 latitude r/>m is represented in fig. 9, by the primitive circle begd, and the angle 

 RMN, by the spherical angle eba; therefore the sine of rmn = the sine of eba 

 = the sine of (ebf + fba + oba) = ap' + a'p + {^'p' — ap) X z, and its co- 

 sine =: a'p' — ap — (ap' -j- ap) X z; which for the sake of brevity may be ex- 

 pressed by I/, and y'; then putting up = n, mn = u, the sine of Mpa = m, its 

 cosine := m', and radius = 1 ; we shall have the sine of mnp expressed by my 

 + my ; and therefore we shall have in the plane triangle m/)N, as the sin, mn/> 



(my' -\- my) : up (n) :: sine m/jn (m) : mn (u); hence u = — r^_ — ;; from which, 



and the equation of the curve (determined above) — = /jn, and consequently the 



duration of the eclipse will become known. 



In prop. 1, the sine of the angle abf is expressed by p -f- p'z, and its cosine 

 by p' — pz, instead of their true values pz' + p'z, and p'z' — pz\ this was done 

 to render the following conclusions more simple than they otherwise would have 

 been ; and as the angle oba is, by hypothesis, bufc small, its cosine will approach 

 so near to the radius, as not to occasion any sensible error in the result ; and the 

 Bame may be observed with regard to what is advanced in prop. 4. 



