ON THE LINES OF CUR VA TURE ON AN ELLIPSOID. 87 



we get 



(& 2 - c 2 ) ud (dvdw) + (c 2 - a*) vd (dwdu) 



+ (a*-tf)wd(dudv)=Q ...... (5). 



Now this is satisfied by the assumptions 



^, -, j- ........... (B), 



f, g, h, being constants. 

 But from (4) we deduce 



du + dv + dw = Q ........................ (6), 



and (B) gives 



du fdu dv div, dv = g du dv dw, dw = hdudv dw. 

 Hence f+g + h = Q ........................ (7), 



which establishes a relation among the otherwise arbitrary con- 

 stants fj g, h. 



Now (B) implies the existence of two linear equations in 

 u, v, w. Hence, a particular solution of (1) is two linear equa- 

 tions connecting the three variables. But the given equation 

 (4) is linear ; hence the solution in question is the one congruent 

 to the problem. 



To find the other relation in u, v, w, eliminate the differen- 

 tials from (3) by means of (B), and there is 



Equations (4) and (5), with the relation (7), contain the com- 

 plete solution of the problem. It is obvious that the apparent 

 want of homogeneity of (B) is wholly immaterial. 



Keeping in mind the values of u, v, w> given by (A), we see 

 that the geometrical interpretation of (8) is, every line of cur- 

 vature on an ellipsoid lies on a conical surface of the second 

 order, of which the vertex is the centre of the ellipsoid. 



To determine the constants, let the line of curvature pass 

 through a point, for which the values of u, v, w, are w t , v^ w^ 

 we have 



(& _ <.) u \ + (<? - ) ^ + (a 2 - 2 ) ^ = 0, 



= 0. 



