of adfelkd Equations • 469 
E X A M P L e nr. 
To find . the value of .x in an equation of the form 
x -f r-x = a. 
Let us take as examples . of this equation x % + 3# 
x' + 3* =.08, and a: 3 -{- 3*' .== . 1 2, which are three of the inftances 
given by Dr. halley, in his Synopfis of the Aftronortsy of 
Comets, to illuftrate the mode of computation that he purfued 
in conftrudting his general table for calculating the place of a. 
comet in a parabolic orbit : and it is obvious, a being put for 
the known fide of the equation, that it may be transformed to 
*s/x xx/s+x'zzs/a: where, if x be confidered as the tangent 
of an arc, the radius of which is \/^, v/ 3 + r* will be the fecaiit 
of that arc and, confequently, by what is (liewn in the firft 
example, an arc mu ft be found, fuch, that the fum of the tabular 
log. fecant and half the tabular log. tangent may be equal to the 
excefs of half the log. of a above £ of the log. of 3. In the 
firft of the above three inftances this excefs will be found, 
18.9431891, in the fecond 19.09.37041, and in the third 
19.1817497; and by running the eye along Gardiner’s 
Tables of Logarithmic Sines and Tangents, it will be found, 
that the firft falls between o° z6 / 2Q n and o° 26' 30", the fe- 
cond between o° 52' 50'' and o° 53' o'\ and the third between 
i° 19' 20" and r° 19' 30" ; and, by purfuing the mode which 
has been defcribed in the two former examples, the fexadt arcs 
will be found 26! 27^, 7,, o° 5-4' 51", 7* and r° 19' 20^, 1, 
and their refpedtive tangents, to the radius s/^, .01333248, 
.0266611, and .0399787, the three values of a fought. And 
in this manner Dr. halley’s table may be extended to any 
length with the utmoft eafe, expedition, and accuracy, 
o 
Thu® 
