—— 

Vol. IV, No. 4.] A General Theory of Osculating Contes. 169 
N.S. 
the equation (6) becomes 
{(X—2)*— Soar i (2pr — 3g*) —2(X—2)(Y—-y) {1 —p4)r 
pq? }+6 {(Y-y)— (aoneye gee o-€7). 
2. As another illustration of the method of last article, we 
may i in general differentials, the equation of the circle 
of curvatur 
h ces of a circle passing through (a, y), (a, y;), is 
evidently of the form (X—#)(X —a#,)+(Y-—y)(Y-y,) =A {(Y¥—-y) 
(a,—2) —(X—2)(y\-y)} 
Therefore the equation of a circle passing through any three 
points, (#, y), (a, 41); (a, yo) is (X—a)(X—2,)+(Y¥-y)(Y—-y) 
(22 —2)(w2— 2) + (yo—y)(ya—y1) 
‘yay j(@1 — 2) — (v9 2)(4)—4) 

{(Y¥—y)(z,-—#)—(X—2) (yij-—y)} (9). 
If now (2, ¥), (@, 41); (ay Yg) be — consecutive points on 
any cnrve, then as in equations (5), #,=a+da, 7,=a+ 2dz + dz, 
with corresponding atheusiae for y; ae Yo. 
erefore, equation (9) gives 
2(dx*+ dy?) . 
2 eee is 
(Xa) Eg ee 

((¥~y)de—(X-2) dy} (10). 
Equation (10) is the equation of the circle of curvature in 
general differentials. Hence, the co-ordinates of the centre of 
curvature and the radius of curvature are given by 
(da® + dy*) dy > 

skid dx dty—dy dx | 
_ , (dat + dy*) da 
oi eae deity = dye r ons 4 

{da + dyt}3 
e dadty—dyda 3} 

Tf x be the independent variable equations (11) become 
xno-G2P yy, ee) 
a 
_ (1+ p22 
q 
. The co-ordinates of the centre of the osculating equilate- 
ral hyperbola oe as ae by differentiating (7) with 
respect to X and Y, 
X=2+ be Cx FD, 
(pr —3q2) 3+ 13). 
ay fpr byt) (14 24) ( 8) 
(pr—3g2)t+ 7 

(12). 

Y=yt+ 
