168 Journal of the Asiatic Society of Bengal. [ April, 1908. 
1. The general Be gc of a conic, apni through two 
given points (a, y) and (2, y,), must be of the 
A(X -#)(X~a,)+ ul Y- cet ti (X-2)(Y—-y) 
p(X ~2)(¥—y)=0 (1). 
as is evident from the number of arbitrary constants involved. 
Therefore, the equilateral hyperbola through (2, y) and 
(2, y,) is of the form 
A {X—2)(X-2#,)-(Y—-y)(Y- hte (X-2)(Y—-y,) 
p(X—a)(Y¥-y)=0 (2), 
maroane ae equilateral hyperbola, through (2, y), (ty Y,)> 
(%g, Y2)s (a 3, Y3)s 
(X—w#)(X—-m)-(Y-y)(Y-y)  (K-#)(Y¥—m)(X-«)(Y—y) 
(wq—2)(a2—21)—(y2—y)(yo—m1) (#2—2)(yo—y,)(e2—m1)(ya-y) [=O (3). 
(ag—#)g—21)—(yg—y)(vs—vi)  (#a—#)(ys—n) (“8-1 )(ys—y) 
or, 
| (X-#)(X— 2) -(Y¥—-y)(¥—y)(X-2)(¥-y) 
(#2—2)(#2— 2) —(y2—y) (y2— 91)(t2— 2 ya— 1) 
1 («3 —@ (23-1) — (48 —y (vs — 1) (73— 2) (ys— vi) 
(Y¥—y)(21 —2) —(X—2)(y)—-y) 
=0 (4) 


(ya—y)(w1 — 2) — (ag—2)(y, —y) 
(yg—y)(#,— 2) —(ag—2)(y,—y) 
if (a, y), (%, Y\)s i Ya), (ag, Y3) be four consecutive 
potas on a curve, then eviden 
2, =e + da, r= Lee @,=2,+dzx, 
Therefore, ana ties dade) = 2+ 2de+d2 2, t,= 5 
+Qdxe+ d*x+ d(a2+ 2dx+ d*xz) =a + 3dz + 3d%z + d3x with cor- (5). 
responding expressions for y), y2, Ys. 
n making substitutions (5) in equation (4), we have, after 
simplifying the determinant by subtracting three times the second 
row from the third and ultimately neglecting all infinitesimals of 
a higher order, 
oS > al (X—2)(Y¥—y) (Y¥—y) de—(X—a) dy 
2ded da —d2ady =O (6). 
re Pha "eid 3( d2ydz + d@zdy) d3y da —d3zdy 
Equation (6) is ag eer: ee ess osculating equilateral 
hyperbola, at any point (z, y) o rve. The coefficients are 
general. go Cae ‘ies tied pope variable being any 
quantity whate 
If the aidependnt variable be x, then d?#=0, d'z=0, and if 
‘we write Pp, q, 7 tor 
dy diy dy 
de’ da®’ da’ 
