" Disquisitiones generales circa Superficies Curvas." 333 



how these new notions are denominated, than to justify their 

 introduction by remarkable and useful theorems to which they 

 give rise. 



The solution of the problem : To find the measure of curva- 

 ture in each point of a curve surface, — presents itself under 

 various forms, according to the manner in which the nature of 

 the curve superficies is given. The simplest manner is, the 

 points in space being expressed by three rectangular co-ordi- 

 nates x, y, z, to represent the one co-ordinate as a function of 

 the two others : in this case the expression which is obtained 

 for the measure of curvature is the simplest. This leads at 

 the same time to a remarkable connection between this mea- 

 sure of curvature and the curvatures of those curves which are 

 produced by the intersection of the curve surface by planes 

 perpendicular to it. It is well known that Euler has first de- 

 monstrated, that two of the intersecting planes, which are like- 

 wise at right angles to each other, have the property, — that to 

 the one belongs the smallest, to the other the greatest radius 

 of curvature ; or rather, that in them the extreme curvatures 

 occur. Now it results from the expression for the measure of 

 curvature just alluded to, that this becomes equal to a fraction 

 whose numerator is unity, and the denominator the product of 

 the two extreme radii of curvature. 



The expression for the measure of curvature becomes less 

 simple when the nature of the curve surface is given by an 

 equation between <r, y, z ; and the former still more compli- 

 cated, when the surface is represented by x, y, z being given as 

 functions of two new variable quantities p, q. In the latter 

 case the expression contains fifteen elements ; viz. the par- 

 tial differential quotients of the first and second order of x, y, z 

 for p and q : but it is less important in itself, than by its being 

 the transition to another expression, which forms one of the 

 most remarkable propositions of this theory. In that manner 

 of representing the nature of the curve surface, the general ex- 

 pression for any linear element of the same, or for \/(dx z -f 

 dy* + dz 2 ) assumes this form: \Z(E,dx 2 +2¥dx ,dy +Gdy 2 ); 

 where E, F, G become functions of p and q ; the above-men- 

 tioned new expression for the measure of curvature contains 

 only these quantities, together with their partial differential 

 quotients of the first and second order. It is evident, there- 

 fore, that for determining the measure of curvature, the ge- 

 neral expression of a linear element only is required, and not 

 the expressions for the co-ordinates x 9 y, z themselves. An 

 immediate consequence of this is the following remarkable 

 theorem : If a curve surface or a portion of the same can be 

 evolved on another plane, the measure of curvature remains 



unchanged 



