106 



such as those of which we may conceive the clouds to be formed, 

 that the suspension of the clouds does not seem to offer any diffi- 

 culty. Had the pressure been strictly equal in all directions in air in 

 the state of motion, the terminal velocity of such globules would 

 have been far larger, and consequently the quantity of water which 

 could have existed in the air in the state of cloud would have been im- 

 mensely diminished. It appears therefore that these small and hi- 

 therto almost unrecognized forces, which depend on internal friction, 

 are essential to the fertility of at least the tropical regions of the 

 earth. 



The author has also applied the theory of internal friction to the 

 calculation of the subsidence of a series of oscillatory waves. On 

 substituting for the index of friction in the resulting formula the nu- 

 merical value deduced from the experiments of Coulomb, it appears 

 that in the long swell of the ocean the effect of friction is insignifi- 

 cant, whereas in the case of the short ripples excited on a small pool 

 by a puff of wind the subsidence due to friction is very rapid. Accord- 

 ingly, short ripples of this kind quickly die away when the breeze 

 that excited them ceases to blow. 



February 24, 1851. 



On some points of the Integral Calculus. By Professor De 

 Morgan. 



Some time ago, Mr. De Morgan communicated to the Society an 

 abstract of some unfinished views on the connexion between the or- 

 dinary and singular solution of a differential equation. The present 

 paper completes those views, and also contains sections on the solu- 

 tion of differential equations by elimination, on the proof of the 

 number of constants which a solution may contain, and on the cri- 

 terion of integrability of a function of x, y, and differential coefficients 

 of y. 



1 . On singular solutions. — As to equations of the first order, the 

 tests obtained in this paper may be described as follows : — 



Mr. De Morgan means by a singular solution any one which is 

 obtained by other process than integration, whether it be contained 

 in the integrated primitive, or not. When the singular solution is 

 not contained in the primitive, he calls it an extraneous solution. 



Let <p(x, y, c) = be the primitive equation, giving c=-$(x, y). 

 The differential equation then is 



and<J»*=:-^ *,= -& 



Every relation between x and y which satisfies either of the fol- 



