ASTRONOMY. 
4 F 7 
For example, Suppofeaftarof the fiiH magnitude fliquld 
have a final! Har of the twelfth magnitude near it; then 
will the partial parallax we are to expect to fee be 
12- 
j 2 X i 
be P=^lp=^p, 
2+—3 
expreffion for which will be 
zmM 
■PyJ i+r*, orZ'PV 
When the ltars are in the pole of the ecliptic, s will be 
“i, and the lalt formula becomes ±p <J z~ ^o-jip. 
Again, let the liars be at home dillance, as 5", from 
each other, and let them be both in the ecliptic. This 
cafe is refolvable into the firft; for, imagine the liar A to 
Hand at I; then the angle A E I may be accounted equal 
^ 
to AO I: and as the foregoing formula, p~— — P , 
gives us the angles A E B, A E C, we are to add A E I 
or 5" to A E B, which will give I KB. In general, let 
the dillance of the liars be d, and let the obferved dillance 
at E be Z>; then will D^dyp, and therefore the whole 
parallax of the annual orbit will be exprelled by ——~ Dd 
Suppofe now the Hars to. differ only in latitude, one be- 
in the ecliptic, the other at feme dillance, as5'*north, when 
iecn at O. This cafe may alio be refolved by the former; 
for imagine the Hars B and C to be elevated at right an¬ 
gles above the plane of the figure, lb that AO B, or A OC, 
may make an angle of 5" at O ; then, initead of the lines 
O A B C, E A, E B, EC, imagine them all to be planes 
at right angles to the figure, and it will appear, that the 
parallax of the liars, in longitude, mull be the fame as if 
the final 1 liar had been without latitude. And, lince the 
Hars B, C, by the motion of the Earth from O to E, will 
not change their latitude, we (hall have the following con- 
HruCtion for finding the dillance of the Hars AB and AC 
at E, and from thence the parallax P. 
Let the triangle ab@ reprefent the fituation of the 
Harsj ab is the fubtenle of 3", the angle under which 
they are fuppofed to be feen at O, The quantity b $ by 
Vol, IT. No. 80. 
m — M. 
or J-J of the total parallax of the larger liar ; and, 
if we fiiould, by obfervation, find the partial parallax be¬ 
tween two Inch liars to amount to 1", then will the total 
parallax P=.±\p— i-Jj.". Again, if the liars be of the 
third and twenty-fourth magnitude, the total parallax will 
fo that, if by obfervation p be 
found to be of a fecond, the whole parallax P will come 
out ^"=0-3428". 
Farther, the Hars being Hill in the ecliptic, fuppofe they 
fhould appear in one line, when the Earth is in feme other 
part of her orbit between E and O; then will the paral¬ 
lax be Hill exprelfed by the fame algebraic formula, and 
one of the maxima will Hill lie at E, the other at O ; but 
the whole effedt will be divided into two parts, which will 
be in proportion to each other, as radius ■— line to radius 
+ « ne ot the flat’s dillance from the nearefl conjunction 
or oppolition. 
When the Hars are any where out of the ecliptic, fitua- 
ted fo as to appear in one line O A B C perpendicular to 
E O, the maximum of parallax will (till be exprelled by 
™ - -P; but there will arife another additional parallax 
mM r 
in the conjunction and oppofition, which will be to that 
which is found ninety degrees before or after the Sun, as 
the line fsj of the latitude of the Hars feen at O, is to ra¬ 
dius (1); and the cffeCt of this parallax will be divided 
into two parts ; half of it lying on one fide of the large 
Har, the other half on the other fide of it. This latter 
parallax will alio be compounded with the former, fo that 
the dillance of the Hars in the conjunction and oppolition 
\\ill then be reprelented by the diagonal of a parallelo¬ 
gram, wliofe lides are the two femi-parallaxes; a general 
m—M 
the former theorem is found ———P, which Is the par- 
mM 
tial parallax, that would have been feen 
by the Earth’s moving from G. to E, if 
both Hars had been in the ecliptic ; but, 
on account of the difference in latitude, 
it will now be reprelented by a ,3, the 
hypothenufe of the triangle a b p : there¬ 
fore in general, putting ab~d , a$—D, 
712 Nl ---- 
we have - ~JD* — d z — P. Hence, 
vi—M 
D being found by obfervation, and. the 
three, d, m, M, given, the total parallax is obtained. 
When the Hars differ in longitude as well as latitude,, 
this cafe may be refolved in the following manner. Let 
the triangle ab (3 reprefent the li- j ^ 
tuation of the liars, ab—d being 
their dillance feen at O, a &—D- 
their dillance feen at E. That the 
change b@, which is produced by 
the Earth’s motion, will be truly 
exprelfed by —— —P, may be pro- 
mM 
ved as before, by fuppofing the 
Har a to have been placed at ot. ct y 
Now, let the angle of poll don baa. be taken by a micro¬ 
meter, or by any other method diffidently ex ad ; then, 
by revolving the triangle aba., we obtain the longitudinal 
and latitudinal differences, aa and b a, of the two Hars. 
Put a as— x, ba—y, and it will be x-\-b$ — aq, whence 
D 
(*•+ 
-M 
mM 
py+f; and 
M—m 
If neither of the Hars fhould be in the ecliptic, nor have 
the fame longitude or latitude, the la.fl theorem will Hill 
ferve to calculate the total parallax, whole maximum will 
lie in E. There‘Will alfo arife another parallax, whofe 
maximum will be in the conjunction and oppolition, which 
will be divided, and iie on different lides of the large Har ; 
but, as the whole parallax is extremely fmall, it is not necel- 
fary to invefirgate every particular cafe of this kind; for, 
by reafon of the divifion of the parallax, which renders 
obfervations taken at any other time, except where it is 
greatefl, very unfavourable, the formulae would be of lit¬ 
tle ufe. Dr. Herfchel doles his account of this theory* 
with a general obfervation on the time and place where 
the maxima of parallax will happen. Thus, when two 
unequal Hars are both in the ecliptic, or, not being in the 
ecliptic, have equal latitudes, north or fouth, and the lar¬ 
ger Har has molt longitude, the maximum of the apparent 
dillance will be when the Sun’s longitude is go? more than 
the Har’s, or when obferved in the morning: and the mi¬ 
nimum, when the longitude of the Sun is 90 0 lefs than 
that of the Har, or when obferved in the evening. But 
when the fmall Har has molt longitude, the maximum and 
minimum, as well as the time of obfervation, will be the¬ 
re verfe of the former. And, when the Hars differ in la¬ 
titude, this makes no alteration in the place of the maxi¬ 
mum or minimum, nor in the time of obfervation ; that is,, 
it is immaterial which of the two Hars has the greater la¬ 
titude. See Phil. Tranf. vol. Ixxii. art. 11. 
Of REFRACTION. 
As one of the principal objeCIs of affronomy is to after- 
tain the fituation of the feveral heavenly bodies, it is ne- 
celfary, as a firh Hep, to underhand the caufes winch occa-i 
lion a falfe appearance of the place of thofe objects, and 
make us fuppofe them in a different fituation.from that 
which they really have. Among thefe caufes refraction 
is to be reckoned. By this term is meant the bending of 
the rays of light as they pals out of one medium into ano¬ 
ther. The Earth is every where fureminded by an hete¬ 
rogeneous fluid, a mixture of air, vapour, and te-rreflriai. 
5 O i exhalations* 
