484 
cos NR’- cos NW cos WR’ | 
s 
cos W= 
sin NW sin WR 
and therefore, 
2sin? NWtan WR’ cosW=2 ee NR'- sn2NW; 
cos WR 
or the expression becomes, 
l sin NW : 
Daan ce 
And sin k”R=sin2 NR =2sin NR cos NR, so that finally 
the expression becomes 
ua sin NW cos NR’ 
~ cos WR’ sin NR cos NR’ 
which proves the theorem. 
‘* To prove the second theorem, take as before , R”, W, 
to denote the angles LRR", LR"R, NWR’, respectively ; 
and moreover, U, U’, U” to denote the angles NUR, NU'R’, 
NU"R", respectively ; then considering, first, the function 
on the left-hand side, viz. : 
sin R”LR' cos RU sin U+sin RLF’ cos R" U" sin UV", 
we have 
sin NR 
sin RU” 
cos RU sin U=sin NR cot RU 
=sin NR cos R cot NR =cos R cos NR, 
and in like manner, 
sin U= 
sin NR” 
sin R"U” 
cos R”U" sin U" = sin NR” cot R”U" 
=sin NR” cos R”cot NR” = cos R” cos NR” = cos R” cos NR, 
and the expression thus becomes 
=cos NFA {sin R”LR' cos R + sin RLR' cos R"}, 
which is 
sin 0” = 
