104 Mr. Woodhouse on the Independence of the 
then, if z represent the arc of a circle, and x the sine, this equa- 
lity* z = x H — yL. + &c. is subject to restrictions, 
for x cannot exceed 1 ; consequently, the greatest value of % that 
can be determined from the equation, must be so determined 
by putting a; ~ i . Let tt = i + m jj m + -p - ^ c • 
Now, from the definition of sine and the nature of the circle, the 
arcs 27 t— z, 67?— z .... (2W+1) S7T — % .... 4^ ~h z •••• Stt — j- .... 
^mr+z, have the same sine ; let these ares be z, z\ &c.. 
and let * + + *c. = X > 
* 3.2 1 5 ,b 
then z' = 27T— X, %' ; == Gtt—X, &c. or generally 
%" ^ j. 27 r — X, or := 4 ^tt 
n any number of the progression o, 1 , 2, 3, 4, &c. 
Or thus, from the conditions contained in the form of the equa- 
tion between z and x, 
since 
V i—z x — £ zS/ * -f- e 
— 1 1 , 
1,2 
-}- &C. 
there is no possible value of z that answers the equation when 
a’ is ~7 1, 
Let / fl= = X + a when z and x begin together, 
V I — X 
a = 0 and % = X 
But the equation 
X ' 
V 1 —x L 
: %' may be derived from x = ( 2 V — 1 ) - “ i 
{ 1 - e~ W ~' }, 
when instead of * is put — z, 6k— z .... (a»+i) s»- z, 
* In the expression * =*+ ~ + -p- + &c - considered abstractedly from its ori- 
gin and application, there is nothing that limits the value of x. Again, by applying 
the operation of-reversion, a: is represented by this form, x i t2i y J *~ 1.2. 3.4.5 
But there is no method, I believe, of proving (I purposely exclude that unproved pro- 
position that every equation has as many roots as dimensions) that instead of z in 
z% . _ Sec. — 0, other quantities, as z', z", &c. may be substituted. 
1.2.3 
x — z-k 
