I 
143 Mr, Ivory on a new Method of deducing 
will fail. This will happen when the compt moves exactly in 
the plane of the ecliptic, in which case r is indeterminate. 
The same thing will likewise happen when the great circle 
which passes through the two extreme places of the comet 
cuts the ecliptic in the points occupied by the earth and the 
sun at the middle observation : for in this case, sin. (e ° — ?i) 
= 0; now since ^cannot become infinitely great, it is necessary 
that I sin. (r° — n) — } vanish together with sin. (e® — w), 
which will make ^ indeterminate. 
The formula for f, which by its sign alone enables us to 
judge of the magnitude of r°, contains the angles n and i: and 
it is necessary to investigate rules for computing those quan- 
tities. By attending to what n and i were made to signify it 
will readily appear that 
let/== 
tan. x' 
tan. X ’ 
tan. i 
then 
tan. X 
tan. x' 
sin. (€—n) sin. (c — n) * 
but 
f sin. (c — n) = sin. (c' — n) : 
• / \ • ( c -4“ € \ c / c c \ * c' 
sin, [c—n] = sin. n) cos. cos. n) sm. - 
c — c 
c'-\-c 
• t f \ • I c' -i-C \ 1/ — C/ I / 1/ -I- 1. t . 
sin. (c —«) =a sin. I “j n) cos. — f- cos. I n) sin. 
therefore by substitution and division we shall get 
tan. — n) =4^ . tan. 
and, if/ = = tan. (45°± ?>) ; then 
tan. (fif + 
— 6 
Z 
g— g 
2 9 
tan. 
= ± tan. flj-' cot. <f. 
To the value of tan. — n), found by this rule, there 
