334 Account of Prof. Gauss's Paper intitled 



unchanged in every point after the evolution. From this fol- 

 lows, as a particular case ; that in a curve surface which can 

 be evolved in a plane, the measure of curvature is always =0. 

 From this is immediately derived the characteristic equation 

 of the surfaces capable of being evolved in a plane; namely, 

 if * be regarded as a function of y and x 



ddz ddz / ddz 



dx* dy* 



(adz \* __ 

 ~dx~d~y) ~ ' 



an equation which, indeed, has long been known, but in the 

 author's opinion has never yet been rigorously demonstrated. 



These propositions lead to a new view of the theory of curve 

 surfaces, and open a wide uncultivated field of investigation. 

 If surfaces be considered not as bounds of bodies, but as bodies 

 one dimension of which is evanescent, and at the same time flexi- 

 ble but not expansible, it will be conceived that two essentially 

 different classes of relations must be distinguished, — those 

 which suppose a definite shape of surface in space, and those 

 which are independent of the various shapes which the surface 

 is capable of assuming. It is of the latter that the author 

 treats ; and it appears by what has been observed above, that 

 the measure of curvature is one of them: but it will easily be 

 seen that the consideration of the figures which can be de- 

 scribed on the surface, their angles, their areas and entire cur- 

 vature, the connection of the points by shortest lines, &c. be- 

 long to this class. All these investigations must proceed from 

 this, That the nature of the curve surface is given by the ex- 

 pression of an indefinite linear element of this form \/(JZdp 2 -h 

 ZFdp.dq + Gdq*). 



The author has inserted in the present paper a part of the 

 investigations on this subject, which have engaged him during 

 several years, confining himself to such as are not too remote 

 from the point where his labours began, and which may serve 

 as general auxiliaries for numerous further investigations. In 

 this notice we must be still shorter, and be content to give 

 only as a specimen the following theorems. — If on a curve sur- 

 face, a system of an infinity of shortest lines, all of equal length, 

 proceed from one point, the line passing through their ex- 

 treme points is at right angles to every one of them. If from 

 every point of any line on a curve surface, shortest lines of equal 

 length and at right angles to that line are drawn, — all these 

 lines are likewise at right angles to that line which connects 

 their other extreme points. Both these theorems, the second 

 of which may be considered as a generalization of the first, are 

 demonstrated analytically, as well as by simple geometrical 

 considerations. The excess of the sum of the angles of a tri- 

 angle 



