328 A. Mukhopadhyay — Bifferential Equation of all Parabolas. [No. 4, 



9H6 (1 + P2) 



r^ + {pr - Sf/f = 



(2Ha7 + B)^ 



whence 



But since 



r2 + (pr - 322)2 I 2 27(1 + P^)' 

 ¥-% f\/a-g\/b 



2-6^ 



P __5 - _ !!^ 



we have from 



the relation 



and, therefore 





m = 



5' 

 \/b 



m = 





(a 4- &)-^ 





-2H . 





(1 + P2/ 





54r/ 



which is accordingly the formula sought. 



Again, let us investigate the coordinates of the centre of an equi- 

 lateral hyperbola osculating a curve at a given point. In the first place, 

 we know that in an equilateral hyperbola the projection of tlie radius of 

 curvature at any point on the central radius vector, is equal to that 

 radius vector ; for, if R be the radius vector, 8 the angle between the 

 normal and the radius vector, p the radius of curvature, and a the semi- 

 axis-transverse, we can easily show that 



P = - ^ . cos 8 = -, , 



whence 



R = — p cos 8 . 

 Hence, if an equilateral hyperbola osculates a curve at a given point, 

 in the first instance take the tangent and normal at that point as the 

 axes of X and y respectively ; then, expressions for the coordinates of the 

 centre are easily obtained, viz., 



X = R sin 8 , Y = R cos 8 , 

 where R is the distance of the centre from the origin, and 8 the angle 

 between tho central radius vector and normal, so that 



