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ducing this are (in the direction from W to R’). to a point L, 
such that 
cot WL = sees si 
aWwhenwE A: 
and joining VR’, then, 
THEOREM I. 
sin? R” LR’ - sin? RLR 
sin? R’ LR 
a sin VW cos NP’ 
~ cos WR’ sin NR cos NR 
And ifwe suppose also, that an arc 
through N, perpendicular to the base RR”, cuts LR, LR’, 
and LR” produced in the points U, U’, U", then, 
THEOREM II. 
sin R"LR' cos RUsin NUR +sin RLR’ cos R"U" sin NU"R’” 
Hayate cos VW par ani 
=sin R LE WR an NR cosR'U'sin NU'R’. 
‘“‘ The present memoir contains the proof of the two theo- 
rems, and the application of them to the optical theory. 
“To prove the first theorem, I write for shortness 
R, R’, W to denote the angles LRR’, LR’R, NWR’, respec- 
tively ; we have then, 
sin’ k"LR'-sin? RDA’ sin (R’LR'- RLF’) 
sin? kh” LR y sin Rk’ LR 
{sin R’ LR’ cos RLR' - sin RLR’ cos R"LR’} 
1 
~ sink’ LR 
v 1 sin R”Wsm Rk” cos RW-cosLReos LW 
~ sin R’LR sin L W sin LR sin LW 
sn RWsin R cos R”W-cos LR" cos LW 
sin LW sin LR" sin LW 
