485] PROBLEMS AND SOLUTIONS. 
565 
But from the two equations in x, we have 
whence 
therefore 
m 1 2 (sin 2 6 + sin 2 0) = 4c 2 + 2a- 2 + 2b 2 + 4# 2 , m? sin 9 sin </> = 4cx + a 2 — b 2 , 
2x — 
b 2 — a 2 + m 2 sin 6 sin <£> 
2c 
. . n . 4c 2 4- 2b 2 + 2a 2 (b 2 — a 2 + m 2 sin 6 sin 6 
sm 2 6 + sm 2 (f) = — f-' r 
m- 
2 cm 
Hence, comparing the two results, we have 
1 — cos 2 a — cos 2 /3 _ 4c 2 4- 2b 2 + 2d 2 cos a cos /3 b 2 — a 2 
sin 2 a to 2 ’ sin a 
or, as these may also be written, 
m 
2cm ’ Sm “ = 2c ’ 
whence 
m „ , n — b 2 — a 2 n n b 2 — a 2 
sm a = — , cos- a + cos 2 /3 = ——— , 2 cos a cos /3 = 
¿G 
2c 2 ’ 
/ r>\o — a? / — b 2 . m 
(cos a + cos /3)“ = ——, (cos a. — cos p) 2 = —— , sin a = — ; 
C C~“ ZiC 
so that m being put equal to unity, or otherwise assumed at pleasure, a, b, c are 
given functions of a, /3. Or conversely, if a, b, c are assumed at pleasure, then a, /3, m 
are given functions of these quantities. 
o. It is to be remarked that (a, /3) being real, a and b will be imaginary, and 
consequently the points S, H of Professor Sylvester’s theorem are imaginary ( 2 ); if, how 
ever, we write —a 2 , — b 2 in place of a 2 , b 2 respectively, then the radicals \/{(c + x) 2 — a 2 }, 
V{(c — x) 2 — b 2 } have a real geometrical interpretation. The system of relations between 
(a, /3, a, b, c, m) becomes 
/ o b 2 . m 
(cos a + cos p)- = —, (cos a — cos Bf = — , sina = —; 
c 2 c 2 2c 
and considering (a, b, c) as given, we may write 
cos a = 
a + b 
~lic~’ 
cos /3 = 
a — b 
~2c~ 
m — V{4c 2 - (a + 6) 2 }, 
viz. we have either this system or the similar system obtained by writing — b in 
place of b. 
6. Consider two circles with the radii a, b and having the distance of their 
centres = 2c, and to fix the ideas assume that 2c > a + b, that is, that the circles are 
1 Prof. Sylvester remarks that according as /3 is less or greater than a, we may find real values of 
6, (p equal to one another in the one case and supplementary in the other. Hence we must in any case 
be able to make ?’=0 and s = 0 indifferently, which shows a, priori that the line being supposed real, each 
point S, H must be imaginary, but so that the squared distance of either from the line must he a real 
negative quantity, conformably to Prof. Cayley’s important observation in the text. W. J. M.