26 Mr. T. S. Davies on the Foci and Directrices 
First. Tur Exviirse anp HyPErRBota. 
Denote the equations of the ‘given curve and of the line to 
be found, respectively, by 
aytbaytce+dytert+f=0 . . (1) 
pytgqrtr=0 .- « « « « (%) 
Also, let (2%) be the point to be found, 2 the constant ratio, 
and & the given angle of ordination. . 
Retaining the same angular directions of the coordinate- 
axes, transfer the origin to the centre of the curve; then the 
preceding equations become 
ayy t+bryt+ca+f'=0 .. . (3.) 
pytqet+tr=0 . - 6 « « 6 Kh) 
2cd—he Qae—bd 
where a Day P Sra g Ay ee hem (5.) 
, ae + cd? — bde 
faa oe EON + Pin ajre bout ee 
If now D and P represent the respective distances of a point 
in the curve from the point and line to be found, the propo- 
sition affirms that D = m P, or D? = m? P*. Also, if h! # be 
the point in reference to the new axes of coordinates, this con- 
dition will be expressed by 
(a —h'? + 2(~@ — HW) (y — B) cosa + (y — #)? 
_ mm? (py+ gu+r) sin? a ol RIE e 
~ p? —2pqeose+_q* rr. 
It will be convenient, before proceeding further, to lay down 
the following notation. The functions, indeed, from their 
perpetual occurrence in researches concerning the conic sec- 
tions, ought to be designated in some uniform manner. Those 
here employed, from their not interfering with any of the no- 
tations generally used in this class of inquiries, may perhaps 
be found worthy of adoption. 
iS pa 2 GCOS a+ gis 1 ten ey aks 
R? = (a — beosu +c)? + (0? —4ac)sing, . 
se — 0 COs a Oy es en ae ee (8.) 
H = Q—2asin?« = acos2ae—bcosa+e, 
K = Q—- 2csin?« = a — bcosae + € COS 2a, | 
Expand (7.) and identify the result with (3.) by equating 
the homologous coefficients ; then we have 
CSU mn PV sin’ a. sw is) so a le 
b= 2 u? 905 & 12 mi? pg SIN? & ow Wye. sl he ee) 
