482 
ducing this are (in the direction from W to A’) to a point L, 
such that 
i es 
si cae eee ae 
and joining NVA’, then, 
THEOREM I. 
sin? R’ LR’ — sin? RL 
sin? R’ LR 
A sin NW cos NF’ 
~ eos 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 Il. 
sin R’LR’ cos RU sin NUR +sin RLR' cos R”U" sin NU"R” 
Liat cos VW fom, Haye 
=sin R Li Ww sn 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 LRA’, LR'’R, NWR’, respec- 
tively ; we have then, 
sin’? R”LR'-sin? RDA’ | sin (R'LR'- RLRB’) 
sin? R’ LR sin R’LR 
ER {sin R” LR’ cos RLR' - sin RLR’ cos R"LR’} 
1 
~ sin R” 
1 sn R’WsmR” cos RW-cosLR cos LW 
“sm R’LRE| snLwW sin LDR sin LW 
sn RWsin R cos R"W-cos LR" cos LW 
"sn LV sin LR" sin LW 
