494 Rev. B. Bronwin on the Integration and 



to be distinguished from cumarine. On fusing these crystals 

 with hydrate of potassa and testing the product of decompo- 

 sition with sesquichloride of iron, the intense violet colour 

 produced by the salicylic acid which had been formed showed 

 with certainty the presence of cumarine in the original sub- 

 stance. 



The hippuric acid found in the urine of graminivorous 

 animals has in various ways been supposed to be derived from 

 the benzoic acid imagined to exist in the Anthoxanthum odo- 

 ratum. The transformation experienced by that acid in the 

 animal oeconomy appeared in the highest degree to give sup- 

 port to this view. 



It would be desirable to try if the same change takes place 

 with cumarine. The narcotic action of this substance on the 

 animal organism however prevented the making of any direct 

 experiments. 



LXIX. On the Integration and Transformation of certain 

 Differential Equations. By the Rev. Brice Bronwin.* 



r HAVE lately discovered a very simple method of integra- 

 ■*- ting and transforming certain differential equations, which 

 I believe to be quite new. A similar method may be applied 

 to the solution of equations in finite differences. Brief illus- 

 trations of it have been transmitted to the Mathematician and 

 the Cambridge Mathematical Journal, but not yet published. 

 The present paper contains an extension of the method as 

 applied to differential equations. The following subsidiary 

 formulae will be required to facilitate the reductions. And it 



must be observed that D always stands for -j- , and operates 



upon the whole of the quantity which follows it. 



xD n y — 1 D n xy — 7iD n ~ 1 y ~] 



x i j> y — j> ^2 y__2 n D n ~ l xy + n(n—l) T> n ~ 2 y 

 x s Dy = DV y — Sn D n ~ x x*y + 3 n(n — 1 ) D n ~ 2 xy — 



(n-l)(n-2)'D n - 3 y 



These will be verified by actually performing the differen- 

 tiations. 



xT>- n y=T)- n xy + nT)- n - l y 1 «» 



x* D~ w y — D~ n x*y + 2n D""" 1 x y + n{n + 1 )T>- n ~ 2 y J 



&c, which may be verified by successive differentiation. 

 We now proceed to give some examples. 

 * Communicated by the Author. 



(a.) 



