MATHEMATICS AND ASTRONOMY. 
53 
amount of hyperbolic sector of the product, while equation (2) serves to 
determine the plane of the sector. How can the expression in (2) deter 
mine a plane? Compound (fig. 11) cosh A sink B • /3 with cosh B sinh A • a 
and from the extremity P describe a circle with radius sinh A sinh B sin a[i 
in the plane of OP and the perpendicular a/3. The positive tangent OT, 
drawn from 0 to the circle has the direction of the perpendicular to the 
plane. 
This may be readily verified in the case of the product of equal sectors. 
a £ 
Let d = x -f- y • aA 
ß A = x + y - 
then according to the rule for the product in space 
a A ß A = x 2 -f- y* cos aß 
7T 
| X V(. a + /3) + j/—1 y l sin aß • aß | “ 
R 
P 
Fig. 11. 
Fig. 12. 
Suppose that the straight line PB (fig. 12) joining the extremities of 
the arcs is the chord of the product; it is symmetrical with respect to the 
axis a/3. Then 
== i-|/2?/ 2 -f- 2y 2 cos aß = -4 |/l -f- cos aß; 
sinh 
therefore 
therefore by the rule for the plane, which is known to be true, 
cosh a A fi A = ^ (1 -f- cos a/3) -f- 1 -f- ~ (1+ cos a/3), 
= y 1 (1 + cos a/3) 4- 1, 
= y 2 4" 1 4“ y 2 cos a/3, 
= x 2 4* y 1 cos a/3. 
But this last is the value given above by the rule found for space.