[ S 6 7 ] 
refpedtively, and n an invariable quantity ; it is pro- 
pos’d to find, in terms of y and z, the equation of 
which z is a root, and z 2 — 2 x z + 1 = o, a di- 
vifor. 
Taking the fluents of the given fluxionary equa- 
tion, we have, fuppofing x = 1 wheny is = 1, hyp. 
n 
log. of x + V x 2 — i = hyp. log. of y 4* V y z - — 1, or 
x-\- V x 2 — 1 — y + V y z — 1 : Whence, by fubfti 
tuting for x its value 
^ l-i 
•iz 
(found by the equation 
z % — 2 x z + 1 = o), we have z n = y + Vy a — 1 • 
Therefore z n — y is = V y z — 1 ; and, fquaring both 
fides, z zn — 2 y z n -j- y z = y z — 1 . Confequently 
zr n — 2 y z n 1 is = o ; which, fuppofing n a po- 
fitive integer, is the equation fought. 
Now it is obvious, n being fuch an integer, that 
this equation will have as many trinomial divifors, of 
the form z 2 — 2 * s + 1 , as there are values of x cor- 
refponding to a given value of y : Which values of x , 
whenjy is not greater than 1, nor lefs than — 1 (the 
only cafe I purpofe to confider), will not be readily 
n 
obtain’d from the equation x-\-V x z — 1 =y + V > ,a — 1 
found above : But, if we multiply the given fluxion- 
ary equation by we get — 
V — I VI — x 31 Vi — -y* 
of which the equation of the fluents is n x circ. arc 
rad. 1. coflne x = circ. arc rad. 1. coline y j where x 
is = 1 when y is = j, agreeable to the fuppofition we 
made 
