Page:Proceedings of the Royal Society of London Vol 4.djvu/138

Rh 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 equation 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 another 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.