94 
Proceedings of Royal Society of Edinburgh. [sess. 
Note on the Elimination of 6 between tioo Equations of the 
2ndj Degree. 
In the case of the glissette problem one of the equations is of the 1st 
degree. With two general equations of the 2nd degree (a) = 0, 
(&) = 0, and the relation cos^^ + sin^^ = 1, we may form two new 
conjugate equations in the form (6) of the glissette system. To form 
the first of these, is transformed to sin^^ before raising the 
equations to the 3rd degree. The terms of the 3rd degree are then 
eliminated between four equations of the forms,. (&), (c), {d), (e) of 
that system. To form the conjugate equation, sin^^ is transformed 
before operating. A fifth equation is wanted, because we cannot 
use cos^^ + sin^^ = 1 without introducing an additional term. This 
equation is got by eliminating cos0 and the term independent of 9 
between the cubic equations, and then dividing by sin 0, or vice versa. 
Supposing the term of cos^^ transformed, and the equations divided 
by their absolute terms, the last-mentioned equation will be formed 
as under, where X is put for cos^, and Y for sin0 : — 
XW 
XY2 
X2 XY 
Y2 
X Y 
1 
(1) («) 
ai 
h 
Cl c/i 
1 
(2) (5) 
• 
• 
«2 
C 2 C?2 
1 
( 3 ) 
\ 
d. 
-h 
1 
Cl 
= 0 
(4) 
«2 
h 
d^ 
— C?2 
1 
^2 
(l,and 2) 
• 
• 
• {a^ — af) 
(&i ^ 2 ) A ^ 2 ) {d^ - df) 
( 5 ) 
(3, and 4) 
{cyX2 - c^af) {cf2 ~ ^ 2 ^ 1 ) 
• ^ 2 ^ 1 ) 
('^l ^ 2 ) 
• 
( 6 ) 
Subtracting (5) from (6) and dividing by Y, we form 
{cya^ - C 2 «i)X^ + ~ ~ + ^2 “ ^1)^ + (^2 “ + {d^ - df) = 0 
or 
if I gf\x+{h,-i,)Y+ (0 
t 2 1 } V 2 1 
The other equations (formed as above directed) are | 
f V 2 - ^2<^iWy+ J Y2 + (a2-a3)X + (5i-i2)Y+ / “i'’2l=0(c?) I 
\ + a^d^ - a.gd^ j \+ a^c^ - a^c2 j - a^c^ f | 
«iC2 - } XY + i 1 Y2 + {b^ - b^)X + j ^ 1^2 “ I Y -f I =0 {e)tf 
-J- b-^d 2 — b^d -^ ) \ + ~ ^1^2 J ( + ^2 ) I “ ^2^1 J I 
