of Unhwim Quantities. 
The values of x and ?/ deduced from these, will satisfy the ori- 
ginal equation, and the same values, when substituted for x and 
e/ in any other equation, will produce an equation, from which 
— may be found, thence x' and ?/', also x and y may be found. 
s 
4. Equations of the third order, involving two unknown 
quantities, may, in certain cases, be satisfied by indeterminate 
functions of x and y ; when such an equation is combined with 
any other equation, the method here explained will apply. 
5. By the analytical artifice here explained, equations which 
involve functions of unknown angles may be transformed into 
common algebraic equations. 
For example, let it be required to determine the angles 
and from the equations, 
m sin. (p=n sin. -vj/, — (1) 
a tan. (p-j-d tan. ^=c ( 2 ) 
In addition to these, the arithmetic of sines furnishes the tw© 
equations, 
Cos. ^ <p + sin.^<p=:l, cos.^ ^^q-sin.^ 
which are satisfied by making 
Cos. 0= 
1-j-r 
2 > 
sin. (p: 
S r 
1 
, 1— .9 
cos. 
Sin. 
1 + 6 * 
hence we find tan. <p 
sin. -ip ^ 
1 -f 
sin. (p £ r 
cos. <p 1— r 
tan. q' 
cos. 4^ 1 — 
By substituting, the equations (1,2) become 
==€ 
The values of r and s may now be found by the usual 
methods. 
Hoyal Military College, 
Fed. 4. 1819. 
