394 ■ -D/'. R. Hey's Propositions concerning 
centres, and H that of the line through the points T, the 
nearer extremities of ST. Let one of the circles have its 
centre in the central line; let F be its mean point, and AO less 
than AF. Let the circles i, ii, and any others, be on the same 
side of AO with B ; and ix &c, on the other. Take any one, 
as circle zi, and draw TH ; also HN, parallel to DT, cutting 
BE in N; and produce HB to Q, making BQ equal to HB. 
Then DT is to DH as DE to DB, as HN or TE to HB, as 
ES to BQ ; whence DT is to DH as DS to DQ, and the 
rectangle SDT or the square of DM to QDH as the square 
of DT to that of DH, therefore as that of HN to that of HB. 
This is * a property of the hyperbola. Therefore all the points 
M are in the curve of an hyperbola, whose transverse axis is 
QH and conjugate axis equal to twice HN, that is, to a diameter 
ST ; having the vertex H, and the-f asymptote BE. 
What follows is an attempt to make such approximation, 
to some law or regularity observed in the directions of the 
axes, as may be of use without constructing the hyperbolic 
curve. J 
As to the remoter circles, it may often be sufficiently ac- 
curate, in practice, to consider their mean points as in a right 
line ; either drawn through one of them, parallel to BE, or 
drawn from F to the mean point of the furthest. So far as 
this can be admitted, the theory already laid down, respecting 
mean points in a right line, is to be taken also as the law or 
laws observed where the centres are so placed. If, in any 
case, one such line be not sufficiently accurate for all the circles 
* T. Newton, prop. VI. f Ib. prop. XXII. 
J The constiuction of prop. VI is to be kept in mind: referred to already in 
prop. IX, X, XI. 
