xlvii 
considered worth while to have briefly indicated the manner 
in which they reproduce themselves in these new processes of 
calculation. 
The vector drawn from the focus first considered to any 
arbitrary point upon the normal, may be represented by the 
expression 
v= (1l—n)a+ nre, (57) 
in which x is an arbitrary scalar; and if this normal intersect 
another normal infinitely near it, then we may write, as the 
expression of this relation, 
0 =dv = (1— 72) da + nedr + (re— a) dn: (58) 
comparing which differential equation with the forms (52) and 
(21) of the differential equation of the surface of revolution 
(15), we can eliminate the scalar differential dn, and deduce 
for n itself the expression . 
Mesa da® 
~ de? + da?” 
(59) 
One way of satisfying these conditions is to suppose 
2 1; dr = 0, pie} (60) 
which comes to considering the intersection of the given nor- 
mal with the axis, and therefore with the other normals from 
points of the same generating circle of the surface of revolu- 
tion : and this intersection is accordingly one centre of curva- 
ture of that surface. The only other way of obtaining an 
intersection of two normals infinitely near, is to suppose, by 
(58), the element da coplanar with a and e, or to pass to con- 
secutive normals contained in the same plane drawn through 
the axis; that is to say, the other centre of curvature of the 
surface is the centre of curvature of its meridian. The length 
of the element of this meridian, that is the length of da, is 
_ denoted by the radical Y (— da®), because the differential da 
is a vector; and the length of the projection of this element 
on the focal vector is + dr = y (+ dr*), because dr is a sea- 
