304 Mr. Baspace on the Influence of Signs 
These values of the chords being employed give 
4 (r° — v sin 6) + 4 (7? — v sin @) = 40 (7° — v2 sin 6°) 
At this step the first of the three stages which have been 
described terminates; the question is now translated into the 
language of Algebra, and must be treated according to its rules: 
the following reductions must then be made 
7? — v? sin 6° + r* — v, sin @ = nr — nv,” sin 0? 
(nv,2 — v2 — v*) sin @ = nr’? — 27°, 
: n— 2 
sm@= +7 / aaa’ 
The second stage is here concluded by the solution of the equation 
to which the first conducted us, and we have now to explain the 
meaning of its two roots, and the modifications which may arise 
from any peculiar relations amongst the data. 
The two signs signify that the angle @, may be measured 
either above the diameter or below it, as is apparent on inspect- 
ing the figure. As the result contains an even radical, we must 
enquire if in all cases a solution is possible, and if not, what are 
the conditions of possibility. For this purpose we observe that 
the numerator and denominator must be both positive or both 
negative, consequently 
nm —2>0, and nv, — v; — v° > 0, 
or 
n— 2<0, and nv,2 — v7 — v <0, and also sin @ <1, 
are the two sets of conditions; in all other cases the question is 
impossible. 
From this, however, must be excepted the case of the nume- 
rator and denominator simultaneously vanishing in consequence 
of the following relations taking place amongst the data, 
n—2=0, and nv,* — v1 — v = 0, 
