114 



determinate form will affect the equations of condition. As by this 

 supposition, these are reduced to equations from which would result 

 the conditions that would render all the coefficients of the determi- 

 ning equation 0, it is inferred that ^ must be indeterminate, and 



that therefore, at an umbilic there issue lines of curvature in all di- 

 rections. 



Of these lines of curvature, it is possible that some may be di- 

 stinguished from others, by proceeding from the point in more inti- 

 mate contact mth the osculating sphere, and it is therefore necessary 

 to determine the analytical character of such particular lines of cur- 

 vature. With this view, the author resumes the equation of the 

 normal in the immediate vicinity of the umbilic. He then points out, 

 that a straight line, whose equations contain the second differential 

 coefficients, thus involving a new condition, will coincide more 

 nearly with this normal, than can any straight line not having that 

 condition. That the lines may intersect in the centre of the oscu- 

 lating sphere, their equations must simultaneously exist ; and thus, 

 that which most nearly coincides with the normad in the immediate 

 vicinity of the umbilic has the new conditions, 



d'A d'-A dj d^ dy^ _ 

 dx^ ^ dx dy' Jx'^ dy'^ 'dx^'~^' 



1^' '^^dxdy' dx d^ ' 'dx'~^' 



in addition to the former ones. 



From this it appears, that when the direction of a line of curvature 

 issuing from an umbilic is such as to fulfil, besides the ordinary con- 

 ditions, the foregoing new conditions, that line of cur^'ature will lie 

 more closely to the osculating sphere than any other not satisfying 

 these additional equations. These new conditions arise from differ- 

 entiating the preceding ones with respect to x and y, considered as 

 d y 



dependent, regarding ^ as constant ; and as these are equivalent 



to a single condition (Monge's and Dupin's equation) it will be suf- 

 ficient to differentiate this, under the above restrictions, in order to 

 obtain a single condition equivalent to the new ones. As this single 

 condition will apj>€ar under the form of an equation of the third 



degree in — , there will, in general, be at least one line of curvature, 



dx 



proceeding from the umbilic, of more than ordinary closeness to the 

 osculating sphere ; and there may be three. If, indeed, this equa- 

 tion of the third degree should, hie that of the second from which 

 it is deduced, be identical for the coordinates of the umbilic, it is 

 obvious from the investigation, that we must then proceed to an- 

 other differentiation ; and so on, till we arrive at a determinate 

 equation, the real roots of which will make known the number and 

 directions of the lines of closest contact. 



