394 Intelligence and Miscellaneous Articles. 



results between x, y, v. This last equation is often more manageable 

 than the original one. 



The process is rendered very simple when the given equation can 

 be reduced to depend on two of the form 



p=<p(x,y, v) q=ty(x,y,v). 



The second method was completed, Mr. De Morgan states, and out 

 of his hands for transmission to the Society, when he discovered that 

 Mouge had communicated it to the Institute, by which body it was 

 never published. But M. Chasles found it among the manuscripts 

 of the Institute, and stated it a few years ago in one of the notes to 

 his Apercu Historique .... des Me'thodes en Geometric Its occurrence 

 in the voluminous additions made to a work which itself treats only 

 of geometry, seems to have prevented it from becoming known to 

 any writer on the differential calculus. Certain particular cases 

 appear in the writings of Legendre and Lacroix. 



Let the equation be <p(x,y, z,p, q, r, s, t)=0. Change x into p, 



y into q, z into px + qy—z, p into x, q into y, r into , s into 



— — — , t into If the equation thus resulting can be inte- 



rt— s 2 rt— s 2 



grated, let its solution be Z=\J/(X, Y). Then the solution of the 



original equation can be obtained by eliminating X, Y, Z from 



Z=*(X,Y) *=g y=g- z=xX+yY-Z. 



In both methods the most effective mode of proceeding is to find 

 what Lagrange calls a primary solution, containing two arbitrary con- 

 stants, and then to use that primary solution. 



LXII. Intelligence and Miscellaneous Articles. 



ON THE RIPENING OF FRUITS AND THE GELATINOUS BODIES 

 OF VEGETABLES. BY M. E. FREMY. 



THE author gives the following summary of the facts detailed in 

 his memoir on the above-named subjects : — 



1 . There exists in the tissues of vegetables, and principally in the 

 pulps of fruits and of roots, a substance insoluble in water, which he 

 has named pectose ; its characteristic property is that of being con- 

 verted into pectin, by the influence of the weakest acids. It differs 

 essentially from cellulose in all its properties. 



2. Pectin exists in the juices of ripe fruits ; it may be artificially 

 obtained by causing boiling weakly acid liquors to act upon pectose. 

 Pectin ought to be considered as a weak acid ; it does not precipitate 

 the neutral acetate of lead, and changes into pectic acid under the 

 influence of soluble bases. 



3. Pectin, submitted for some time to the action of boiling water, 

 acquires the property of precipitating neutral acetate of lead, and is 

 converted into a new substance which M. Fremy calls parapectin ; 



