452 Mr MAXWELL, ON THE 



the two systems of lines. One case is when the lines of the first system are geodesic, or 

 " shortest" lines having their origin in a point, and the second system is drawn so as to cut 

 off equal lengths from the curves of the first system. 



The angle which the tangent at the origin of a line of the first system makes with a fixed 

 line is taken as one of the co-ordinates, and the distance of the point measured along that line 

 as the other. 



It is shewn that the two systems intersect at right angles, and a simple expression is found 

 for the specific curvature at any point. 



M. Liouville {Journal, Tom. XII.) has adopted a different mode of simplifying the pro- 

 blem. He has shewn that on every surface it is possible to find two systems of curves inter- 

 secting at right angles, such that the length and breadth of every element into which the 

 surface is thus divided shall be equal, and that an infinite number of such systems may be 

 found. By means of these curves he has found a much simpler expression for the specific 

 curvature than that given by M. Gauss. 



He has also given, in a note to his edition of Monge, a method of testing two given 

 surfaces in order to determine whether they are applicable to one another. He first draws on 

 both surfaces lines of equal specific curvature, and determines the distance between two corre- 

 sponding consecutive lines of curvature in both surfaces. 



If by assuming the origin properly these distances can be made equal for every part of 

 the surface, the two surfaces can be applied to each other. He has developed the theorem 

 analytically, of which this is only the geometrical interpretation. 



When the lines of equal specific curvature are equidistant throughout their whole length, 

 as in the case of surfaces of revolution, the surfaces may be applied to one another in an infinite 

 variety of ways. 



When the specific curvature at every point of the surface is positive and equal to a 2 , the 

 surface may be applied to a sphere of radius a, and when the specific curvature is negative 

 = — a 8 it may be applied to the surface of revolution which cuts at right angles all the spheres 

 of radius a, and whose centres are in a straight line. 



M. Bertrand has given in the Xlllth Vol. of Liouville's Journal a very simple and ele- 

 gant proof of the theorem of M. Gauss about the product of the radii of curvature. 



He supposes one extremity of an inextensible thread to be fixed at a point in the surface, 

 and a closed curve to be described on the surface by the other extremity, the thread being 

 stretched all the while. It is evident that the length of such a curve cannot be altered by 

 bending the surface. He then calculates the length of this curve, considering the length of 

 the thread small, and finds that it depends on the product of the principal radii of curvature of 

 the surface at the fixed point. His memoir is followed by a note of M. Diguet, who deduces 

 the same result from a consideration of the area of the same curve ; and by an independent 

 memoir of M. Puiseux, who seems to give the same proof at somewhat greater length. 



Note. Since this paper was written, I have seen the Rev. Professor Jellett's Memoir, On the 

 Properties of Inextensible Surfaces. It is to be found in the Transactions of the Royal Irish Aca- 

 demy, Vol. XXII. Science, &c, and was read May 23, 1853. Professor Jellett has obtained a system 



