43 
1892-93.] Prof. Anglin on Properties of the Parabola. 
(3.) Making the polar of {pc'^y') identical with the equation to the 
directrix, we have 
a{ax' + (By) +g ^ p{ax' + Py) +/ 
2(^ - a COS w) - 2(a - P COS co) 
^ +fy+(^) 
f^ + 9^ - ’^fy cos (0 - ’ 
each of which ratios = {fBg - a/)/2y^. 
Thus X and ij\ the co-ordinates of the focus, are given by 
{ax -h ^y)f ■\-ag^^f- (a/ 4- ^g) cos <o = 0 ) , 
+/^ + - V9 cos w cy2 = 0 i ' 
and hence 
2(a/-/S(?)y2.!j; 
= /3(?^ - /^) + + 2a/(/cos a-g), 
2 («/- Pg)f-v 
= “0^ -P) - - 2^? cos 0 ) -/) . 
III. When the curve is referred to a pair of tangents as axes. 
If «, b be the lengths of the axes, the equation to the curve may 
be written 
_ y\ 
\a b) 
^x 
a 
0 , 
so that all the results may either be at once deduced from those 
previously obtained for oblique axes, or they may be found inde- 
pendently in like manner. It is, however, desirable to investigate 
the results in this case by peculiar methods, on account of certain 
tangent properties of the curve. 
Taking the usual figure, complete the parallelogram QOQ'W. 
Then the triangles OSQ', OQW are equiangular. 
.-. OS.OW = OQ.OQ'. 
Also, drawing SR parallel to OQ, the triangle ORS is equiangular 
to each of the triangles OSQ, OSQ' \ 
.-. OR.OQ' = OS^ = SR.OQ. 
Further, since ^MOM' = 2^QOQ', the .^MOV = ^QOQ'. 
Hence, if OQ', OQ be axes of x and y respectively, and 
-/QOQ' = 0 ), we have 
ax = by = 0^^ = a%‘^/e^, 
