316 
ON THE EXPRESSION OF THE COORDINATES OF A POINT OF A 
[606 
viz. writing for a moment a# + & = 2 Vb~ — ac . u, this is 
Hence, assuming 
4 u 3 — 3 u + 
a 2 d — 3 cibc + 2 b 3 
2 (b 2 — ac) Vb 2 — ac 
= 0. 
— COS <f) = 
a 2 d — 3 abc + 2 b 3 
2 (br — ac) Vi 2 — ac’ 
then we have 4<u 3 — 3u — cos </> = 0; consequently u has the three values cos ^<j>, cos 27t), 
cos £ ((f) — 2tt), and we may regard cos as representing any one of these values. 
We have thus ax + b= 2*Jb 2 - ac cos ^(f), and y = Xx, giving x and y as functions 
of X and <j), that is, of X. But for their expression in this manner we introduce the 
irrationality V& 2 — ac, which is of the form V(l, X) 6 , and the trisection or derivation 
of cos from a given value of cos </>; viz. we have, as above, — cos 0, a function of 
X of the form 
(l, xy+(i, X)«V(1, xy. 
The equation for (j) may be expressed in the equivalent forms 
sin (f> = 
— tan cf) = 
and inasmuch as we have 
a V— (a 2 d 2 + 4ac 3 + 4b 3 d — 6abcd — 3b 2 c 2 ) 
(b 2 — ac) Vb 2 — ac 
a V— (a 2 d 2 + 4ac 3 + 4b 3 d — 6abed — 3b 2 c 2 ) 
ard — 3 abc + 2 b 3 
we may, instead of 
write 
2 Vfr 2 — ac 
a 2 d — 3 abc + 2 b 3 
(b 2 — ac) cos </> ’ 
ax 
+ b = 2 V& 2 — ac cos $(f), 
h _ (a 2 d — 3abc + 2b 3 ) cos ^<f) 
ax -f- o — 7," r , 
(b- — ac) cos 9 
or, what is the same thing, 
— (a 2 d — 3 abc + 2b 3 ) 
(b 2 — ac) (4 cos¡ 2 — 3) ‘ 
The formulae may be simplified by introducing y, a function of X, determined by 
the equation 
c/m 2 — 2 b/i + a = 0 ; 
viz. this equation is 
i(l, X)V-f(l, *)> + (!, ^) 4 = 0, 
so that (X, y,) may be regarded as coordinates of a point on a nodal quartic curve, 
or a quartic curve of the next inferior deficiency 2. And we then have 
(c/m — b) = V& 2 — ac,