250 Proceedings of Royal Society of Edinburgh. [sess. 
He then takes each of these nine equations along with the one 
from which it was derived, and by subtraction obtains nine new 
equations, which he groups as follows : — 
0 = (L — A) * {fi'y" — fi"y') + (L cos v — c )’{y , a — y"a) + (Lcos fi — b)‘ {a' fi" — a'fi') 
0; = (L COS V - C ) * {fi'y" - fi"y') + - y'a) + (L COS A - Cb) ’(a'/3" - a'fi') 
0 = (L cos /a — b ) * {fi'y" — fi'y') + (L cos A — a) * (y'a" — y "a) + (L — C)'(a'fi" — a"fi') 
0 = (M - A)’ {fi"y — fiy") + (M cos v - c ) '(y'a - y a") + (L cos jx - b ) •(a , /8 - afi") 
0 = (M cos v - c ) • (£"7 - $y") + (M-B)*( 7 "a - ya" ) + (M cos A - a)’ (a" 13 - a/ 3 ") 
0 = (M cos /a - b ) * (£"7 - £ 7 ") + (M cos A - a) * (y'a - 7 a") + M - C) * (a"j 8 - aj 8 ") 
0 = (N — A) ' (fiy' — fi'y) + (N COS v — C ) ’ {ya! — y'a) + (N COS jx — b)' ( afi’ — a fi) 
0 = (N cos v - C ) '{fiy - fi'y) + (N-B)'( 7 a' - y'a) + (N cos A - a)’ {afi' - a' fi) 
0 = (IT COS fx — b ) ‘ {fiy' — fi'y) + (N COS A — CL )’{ya — y'a) + (IT — C )' {afi! — afi), 
How from the first of these groups of three it is possible to 
eliminate /3'y" - fd"y , y'a" - y'a , a! ft" - a" ft' ; from the second, 
ft"y — ft'y " , y 'a — ya" , a" ft -aft"; and from the third, fty — ft'y , 
ya - y'a , aft' - aft ; and this being done there is obtained the set 
of three equations 
0 = (L - A)(L - B)(L - C) + 2(L cos A - a )( L cos g - b )( L cos v - c) 
- (L - A)(L cos A - a) 2 - (L - B)(L cos g-b) 2 - (L- C)(L cos v - c) 2 , 
0 = (M - A)(M - B)(M - C) + 2(M cos A - a )( M cos g - b )( M cos v- c) 
- (M - A)(McosA - a) 2 - (M - B)(Mco Sy a - b) 2 - (M - C)(M cosv - c)\ 
0 = (H - A)(H - B)(H - C) + 2(H cos A - a)(H cos g - 6)(N cos v - c) 
- (H - A)(H cos A - a) 2 - (N - B)(Hcos/x - b) 2 - (N - C)(Ncos v-c) 2 ; 
from which it is clear that the unknowns L , M , H are the three 
roots of the equation in x , 
0 = (x - A)(x - B)(x - C) + 2(x cos A - a)(x cos g - 6)(x cos v-c) 
- (x - A)(x cos A - a) 2 - (x - B)(x cos g - b) 2 - (x - C)(x cos v - c) 2 , 
and therefore may be considered as expressible in terms of the 
nine knowns, A,B,C,a,5,c,A,/x,v. 
To obtain the remaining unknowns — which, be it noted, are not 
a 1 ft > 7 
' O' ’ 
a , p , y 
a-", F, y" 
P'y" - fi'y > 
P"y - Py " . 
Py - P’y . 
t rr rr / 
ya -ya, 
y'a — ya , 
ya - ya , 
aft" — a" ft ' , 
a" ft — aft" 
aft' — aft , 
but 
