[ 3i ] 
Let the fine of EBFm#, its cofine—z/, the tan- 
gent of BF=z/3, its cofine —b' , the fine of F Ba=p, 
its co-line—^/, the fine of zzBA — z, the fine of 
AB— Z, its cofine = Z, the fine of PAO — and 
radius = i ; then we fhall have the fine of ABF 
= the fine of FB^z-J- aBA ■=zp^-p'z i and its cofine 
— ft — pz: Therefore by trigonometry we final! have 
in the right angled fpherical triangle ABF, as Rad. 
( 1 ) : cofine BF (b') :: fine ABF [p-\-p'z : cofine 
BAF — fine of LAB, or its equal . PAO ; therefore 
£ = b' x/> -\-p'z — tlie fine of the required inclina- 
tion of the planes LPH, BOD. In like manner in 
the fame triangle it will be as rad. (i) : cotan. BF 
( j ) : : cofine ABF (p' — pz) : cotan. B A = § ; 
hence Z = B > and Z == — , which 
are the fine and cofine of the required arc AB. 
COROLLARY I. 
If inftead of a fphere we now fuppofe BEGK re- 
prefents a prolate fpheroid, whofe axis is CP; the 
figures of the fedtions LPH, BOD, Sec. inftead of 
circles, will become ellipfes (by the lemma) ; but it 
is evident that the inclinations of thofe planes to 
each other, and likewife the inclination of the right 
lines AC, BC, or the angle ACB, will remain un- 
altered. 
COROLLARY II. 
If BEGK reprefents any primary planet revolving 
about the fun in an orbit whofe plane co-incides 
with 
