. [ 37 1 
nature of the required curve; but the figure of the 
fun is fo nearly fpherical, that it was not thought 
necefiary to embarrafs the folution with that confi- 
deration. 
Hence the duration of an eclipfe of a given fa- 
telles may be determined in the following manner : 
Let BRC (fig. 6.) be the fedtion of the fhadow, 
through which the fatelles paffes, N/>N the path 
of the fatelles, making the given angle N p M, with 
the circle of latitude Ry> M ; BMC a part of the 
primary’s orbit produced, and M p the given latitude 
of the fatelles at the time of the fyzygia ; the circle of 
latitude R/>M is reprefented in fig. i. by the primitive 
circle B EGD, and the angle R iVL N, by the fpherF 
cal angle E B A ; therefore the fine of R M N = the 
fine of E B A = the fine of E B F F B a -f- a B A 
= ap r -f- a' p -\- a' p' — ap x z, and its cofine 
a= a'p' —ap — ap r + a'pX.Z', which for the fake of 
brevity may be exprefifed by y, and y ' ; then putting 
M/> — n, M N = u, the fine of Mp N = m, its co- 
fine = m\ and radius = i ; we lhall have the fine 
of M N P exprefi: by my' -f- m'y ; and therefore we 
fhall have in the plane triangle M/>N, as the fin.MN/? 
{rn y -j -my) : M p (yi) : : fine My> N (/;/) : M N ( v ) ; 
hence v — _j ; from which,, and the equation. 
of the curve (determined above) ~ ™ p N, and con- 
fequently, the duration of the eclipfe will become- 
known. 
In prop. I. the fine of the angle ABF is expreficd 
by p - \~p'z } and its cofine by p' —p z, inftead of their 
true, values pz' -\-p'z> and p'z' — pz ; this was done. 
