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 = {1 — n) a + nn, (.57) 



in which n is an arbitrary scalar ; and if this normal intersect 

 another normal infinitely near it, then we may write, as the 

 expression of this relation, 



=dv = {l~-n)da + mdr + (re — a)dn: (58) 

 comparing which differential equation with the forms (52) and 

 (21) of the difi'erential equation of the surface of revolution 

 (15), we can eliminate the scalar difi'erential dn, and deduce 

 for n itself the expression 



"=cl7^- (59) 



One way of satisfying these conditions is to suppose 



nzz 1, d;- = 0, v = re ; (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 V {— du-^), because the differential da 

 is a vector; and the length of the projection of this element 

 on the focal vector is ± dr =. \J (+ dr"'), because dr is a sea- 



