136 



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

 "Application de l'Analyse a, la Geometrie," edit. 1850. 



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

 Liouville's Journal, 1847. 



3. " Demonstration d'un Theoreme 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 meme The'oreme," parM. Puiseux. — Liouville's Jour- 

 nal, 1848. 



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



March 28, 1854. 



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 Senarmont, 



o 



Wiedemann and Angstrom. 



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

 theory of differential equations. 



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

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



JTT 



The symbol for a differential coefficient, \J X for — , &c, long used 



dx 



by the author, is thus extended. By V x \ Piq is meant d\J : dx with 



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



U x \ p , q means U^+U^ + U^; and \J x \ y means U x +XJ y y'. 



Differentiations are sometimes expressed thus : ^11 = 17^, dx, 

 d x , pU = XJ x dx + XJ y dy . 



When it is only requisite to express functional relation, without 

 specification of form, (x, y,z) = 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 u=u(x, y, z) may signify that u 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 

 which destroy the effect of its variation upon the form of differential 

 coefficients, it is called self -compensating . Thus f(x, y, a) = 0, 

 <p a (x,y,a)=0, contain the self-compensating variable a. Similarly, 

 when <p(x,y, a,b) = is accompanied by f a da + (p b db=0, a and b are 

 mutually compensative, and primordinally. The addition of 



<t>a{x\y) da + fi,(s\ y ) db = 



makes a and b biordinally compensative. 



3. When a finite change in x makes an infinite change in y, it 

 makes an infinite change in y' : y, in y" : y' , &c. When either or 



