■ Dr, 'Waring on 
15. When the given equations A = o, B=o, C = o, &c._ 
contain Irrational or other quantities whofe fluxions can be 
found ; and approximates ( 'a , b 9 c , d, &cf) are .given to each ot 
the unknown J)h z 9 ^9 &c.^ contained in the 
gjEvcn equations ; let b~\-y / ~y 9 c- 3 r z / ~z 9 d + v' — Vy 
,&c. fuhftitufe in the quantities .A,,^ (—) 9 C*~) * 
&c - {4)-» (4) » (4), (4 ),&c.;(4t)» (4)» &c - for 
x .*• v ' xyz' x y' 
y 9 %>, Vy &c. refpe&ively b 9 c 9 d 9 &c. ; let the refulting cor- 
cefpondent values be A 7 , /'A* ) 5 ('—) > &c.whence may be de- 
X X ' X j ' 
duced the equation A / + / ) x' + 4 * + 
a , 3' , + (AI)^V + &c.)+&c. = o in which no irrational, &c. 
x XZ ' 
function of the approximates x'y y' 9 %' , &c. is contained ; and 
in the fame manner may the remaining equations B = o, C = o, 
&c. be transformed into others, in which no irrational func- 
tion of the approximates (# 7 , y f , &c.) is contained, and 
from the refulting equations may be found approximate values 
of the quantities x' \y\ %'y &c. 
If there be given only two equations A~o and B~o con- 
taining two unknown qualities# andy, and all the quantities of 
the refulting equations, in which are contained more than one 
dimeniion of the quantities x / and y r be rejected, there will re- 
fult the two equations A'+j^AJI x' 4- y / = o and B 7 + ] 
x y 
J x '+ ( —) / *= °> from which may be found x' an Ay', the 
fame 
