450 Cambridge Philosophical Society. 



place, after which the system of lines must be altered that the bend- 

 ing may continue. Such lines of bending are in continual motion 

 over the surface during bending, and may be called " instantaneous 

 lines of bending." When a second condition is satisfied, a finite 

 amount of bending may take place about the same system of lines. 

 Such a system may be called a " permanent system of lines of 

 bending." 



Every conception required by the problem is thus rendered per- 

 fectly definite and intelligible, and the difficulties of further investi- 

 gation are entirely analytical. No attempt has been made to over- 

 come these, as the elementary considerations pre^aously employed 

 would soon become too complicated to be of any use. 



For the analytical treatment of the subject the reader is referred 

 to the following memoirs : — 



1. " Disquisitiones generales circa superficies curvas," by M, C. 

 F. Gauss (1827). — Comm. Recentiores Gott. vol. vi.; andinMonge's 

 "Application de I'Analyse a la Geometric," edit. 1850. 



2. " Sur un Theoreme de M. Gauss, &c.," par J. Liouville. — 

 Liouville's Journal, 1847. 



3. " Demonstration d'un Th6orfeme de M. Gauss," par M. J. Ber- 

 trand. — Liouville's Journal, 1848. 



4. " Demonstration d'un Theoreme," Note de M. Diguet. — Liou- 

 ville's Journal, 1848. 



5. " Sur le raeme Theoreme," parM. Paiseux.— Liouville's Jour- 

 nal, 1848. 



And two notes appended by M. Liouville to his edition of Monge. 



March 28. — Prof. Miller gave an account of the relation between 

 the physical characters and form of crystals of the oblique system as 

 established by the observations of Mitscherlich, Neumann, De Se- 



narmont, Wiedemann and Angstrom. 



A paper was read by Prof. De Morgan on some Points in the 

 theory of differential equations. 



1. The -words primordinal, hiordinal, &c. are used in abbreviation 

 of the phrases ' of the first order,' ' of the second order,' &c. 



The symbol for a diflferential coefficient, U^. for — , &c., long used 



dx 



by the author, is thus extended. By U,,. i^, , is meant rfU ; dx with 



reference to x as contained in p and q, as well as explicitly. Thus 



U,|p,, means U,.-j-Upjo^+U^g^ ; and U^-ij, means U,,+ U^y'. 



Differentiations are sometimes expressed thus : d^\] ■= U,. dx, 

 dx, 3/U = \jjlx -\-\]ydy. 



When it is only requisite to express functional relation, without 

 specification of form, {x, y,z)-=0 or z'={x,y) may signify an equa- 

 tion between x, y, and z. A letter may be used as its own functional 

 symbol : thus w = a(,r, y, z) may signify that « is a function of x, y, z. 

 And in ' for u write u(x,y, z) ' there is a convenient abbreviation of 

 ' for u substitute its value in terms of x, y, z.' 



2. When, as so often happens, a variable enters under relations 



