9% Mr „ Woodhouse on the Independence of the 
— -v'- 
4 r - 
■) 
• 3 • 2-1 { * + r* v=I } { }- (/ v=r _ r -w-i). 
2 — I 
- r / 
— V — X * 
2” 
£ — £ 
.£v: 
z n 
r, y 
2 . 2— l (AV— l ^. £ — ,2.2- 
or, generally, 
= 2”. 2-' [;l' /=r l+ }. 2~l{ ^ V '“+ } . 
(,?vzr^-jv=rj.... 
. . a — *( f " ~- x '~ ! _j_ ~ A * } • (2^ — 1)' 
3r 1? A 
Which is the analytical translation of sin. x= 2*. cos. ~ . cos.~ &c> 
Euler, and after him other authors, have demonstrated these 
formulas by the aid of logarithms, and of theorems drawn from 
geometry. 
X. Euler and Lagrange have treated certain differential 
equations, which are said to admit for their complete integration 
an algebraic form, although the integration of each particular 
term depends on the quadrature of the circle and hyperbola. I 
purpose to integrate these differential equations, by the method 
adopted in Articles V. VI. 
Let fx>fy, denote functions of x and y. 
Suppose the differential equation to be 
£1 4. Z. = 0 ; then fx +fy = a when x = e-^,y= z f K Hence, 
xy sa efx+fy = e a = A, a constant quantity, 
sdly. Let 7==- + = 0 
••• fx +fy— a < x b ^ n g = { 2v/ — 1 }~ K {f’ ¥ -' —r f " ¥ ~‘)> 
and y = 2 >/”)—. — or = 2—. 
i-fi'/—*), and \/ ] — d’— 2—. '). 
Hence, .r. v^ 1 — d ) +d v / ( 1 ~- c *) 
— ( 2 | £ {/’+/>W - • — f —( f*+JHW-' | 
-av'— -S 
| = A, a constant quantity. 
