y 4 Dr, W a r i n g’s Obfervatio ns 
111 fome cafes the impoffible part may vanilh, which may 
be the cafe in a quantity of the following formula, viz, 
\Z a + ay/ - b + V a + / 3 \Z — b + \ ) /a-\-yxy~b + &c. where a> 
(2, y, Sec. denote the 2 rn roots of X/ ~ 1 • The general princi- 
ples of difeovering the cafes in which this happens have been 
given in the Meditationes Algebraical. 
The roots of the equation ^±1=0 will be found from 
common algebra and thefe principles, if h is not greater than 
10; or, more generally, if h — 7} x ^ x 4/' . . io /3 , where /, 
denote any whole numbers:: or, in general, the 
roots of the above-mentioned equation, or even of the equa- 
tion >v = 7 ±L±Mv/-i, can be found from tables of fines. 
The fame principles may be applied to the difeovery of the 
values of exponential irrational quantities. 
In the Mifcel. Analy. was given, from a fubflitution 
invented by me and not fimilar to any before given, a refolu- 
tion of equations, which contains the refolutions of all equa- 
tions before given, and from which the refolutions of fome 
equations, not before delivered, have been added. 
Part II. 1 . Let an equation A~o involving (r) unknown inde- 
pendent quantities be predicated of another equation containing 
the fame quantities, and the demonibration of it be required. 
lib. Reduce both the equations to equations involving inde- 
pendent quantities only ; then reduce the two equations to 
one, fo that one of the above-mentioned quantities may be ex- 
terminated, and if there refults a felf-evident equation* viz, 
A = A, or A-A=:o, in which the correfpondent terms 
deftroy each other refpedlively ; then the firft equation is julbly 
predicated of the fecond ; that is, if the above-mentioned 
equations 
