578 



four points, a, b, c, d, by another line of the same kind, we 

 shall have 



sin [a, cfl. sin ["6, c1 , , » 



^-^ — li — r-Y — t4 = a constant, 

 sin [a, o\. sin \_c, a\ 



\a, 6] being used to denote the angle between the normals to 

 the surface at the points a and b. 



The normals along the line (A) are all parallel to the 

 tangent plane at the point whose normal coordinates are a 

 and b. Mr. Graves designates this point the pole of the hne. 

 And if a and b be connected by an equation, so that the pole 

 describes some curve of the wth degree, the line (A) will 

 always touch another curve to which n tangent lines of the 

 first degree may in general be drawn from the same point. 

 This relation between the curves being obviously reciprocal, 

 Mr. Graves calls them reciprocal curves. Here is laid the 

 foundation of a theory of polar reciprocals for curves traced 

 upon any given surface. 



Amongst other exemplifications of this method, Mr. 

 Graves employs it to discuss the lines of greatest and least 

 curvature on the surface of an ellipsoid. Their equation in 

 normal coordinates is 



~2 ;,2 c^ 



v2 A. " y2 a- — 



where a^ 6^ c"^ are the squares of the semi-axes of the 

 elHpsoid, and F is indeterminate. 



Now, from the mere fact of this equation being of the 

 second degree, it follows that the sum or difference of the 

 angles between the tangent plane at any point on a line of 

 curvature and two fixed planes is constant. 



But further, all the spherical curves of the second de- 

 gree represented by the preceding equation are biconfocal : 

 and it is easy to show that their common foci are the points 

 on the sphere which correspond to the umbilici of the ellip- 

 soid. Hence, the sum or difference of the angles between 



