92 Mr. Wood house on the Independence of the 
/— r 
J 2 — 
I 
\Z, X 
£ z 
y: 
x'J x - — I 
V — 
Z, X = 
l 
-i £-*y-* | » or V— i (pzV-i i ) ' 
&v — 1 + zy— 1 
And a variety of forms may be obtained, by substituting for x 
different functions of x , in the expression «— ==. Hence, if 
the symbol / is made to denote a reverse operation, the integral 
equations of the preceding differential equations have been 
rightly assigned. All other methods of assigning the integrals, 
by the properties of logarithms, by circular arcs, by logarithmic 
and hyperbolic curves,* are indirect, foreign, and ambiguous. 
VII. An instance or two will shew the advantage of adhering 
to the true and natural derivation of analytical expressions. 
Let x and y be the co-ordinates of a circle; then, 
i = x 1 -f / \ and y = v/ ( l — x* ) , now (arc ) • or ■ *•==*/ (x Vf y '1 
=, in this instance, x • ( i — cc a ) — f : but it has appeared, that 
if x =z }, Z' — x° (i—x a )-l; 
consequently, in a circle, the co-ordinate x, or, in the language 
of trigonometry, the sine x = developement of 
&c 9 
-&c. 
( 2 \/ T- 1 ) 1 . | pV—i ■ C~ zV — 1 } — 
%- 
1.2.3 
-1 — 
.{ £ zl/ ~ 1 + } = 1 — Ti 
+ 
34-5 
2 4 
I. 2.3.4 
and jy or cosine = 2 -1 
1. This method of determining the series for the sine in 
terms of the arc, is, I think, simple, direct, and exact; it requires 
no assumption of a series with indeterminate coefficients, nor 
• By the strange way of determining the meaning and value of analytical expres- 
sions from geometrical considerations, it should seem, as if certain curves were believed 
to have an existence independent of arbitrary appointment. 
