388 Prof. Challis on Integrating Differential 



23. In 1841 Sir David Brewster 37 reproduced some of my 

 results, probably without suspecting that he had been antici- 

 pated, although a full abstract of them was given in a then 

 popular Journal of wide circulation 38 . Brewster formed a film 

 of oil of laurel on water placed in a black vessel, or on the sur- 

 face of diluted or real ink. He justly describes the rings pro- 

 duced as being splendid beyond description. He says : — " These 

 thin plates of oil of laurel exhibit some curious phenomena which 

 I believe have not been noticed. If we wet with water, alcohol, 

 or the oil of laurel itself, the extremity of a short piece of wire, 

 such as a large pin, and hold the pin in the hand so that its 

 head may be above and almost touching the film, the film will 

 recede in little waves of circular shape which form a new system 

 of coloured rings ; and they become covered with the vapour 

 from the fluid on the head of the pin in such small particles that 

 they reflect no light, and the rings appear to be blackened. 

 By withdrawing the pin the film is restored to its former state. 

 The same effect is produced by heating the pin or the fluid upon 

 it to promote evaporation" (p. 51, note). 



24. We now come to the time when Dutrochet published his 

 elaborate essays on what he termed the epipolic force ; and as 

 this forms a kind of middle term between the earlier and the 

 more recent history of this wide subject, we may here conve- 

 niently pause for a time. 



Highgate, N. 

 7th October, 1873. 



XLIX. On Integrating Differential Equations by Factors and Dif- 

 ferentiation, with Applications in the Calculus of Variations. 

 By Professor Challis, M.A.,F.R.S., F.R.A.S.* 



fT^HE method of integrating differential equations discussed 

 A in this communication depends on the following general 

 theorem, which, as far as I am aware, has not been previously 

 enunciated : — 



The differential coefficients ~-> -=-^> -~, &c. being represented 



by P> q> r, &c, let F be a function of oc, y, p, q, r, &c. of any 

 order and degree, and suppose that we have 



37 Phil. Trans. Part I. p. 43. 



88 The Mechanics' Magazine, Sept. 8, 1838, vol. xxix. p. 394. 



* Communicated by the Author. 



