Mr. Herapath on the Solution of Functional Equations. 197 



therefore only remark, that in every unfavourable season, such 

 as the last, the stones are always found larger, relatively to 

 the bulk of the fruit, than in favourable seasons. But of the 

 habits of the trees, or rather of the branches, (for few of the 

 trees have been preserved,) I can speak with much satisfaction. 

 The wood of many has ripened more perfectly, and offers a 

 much stronger and more abundant blossom than is found on 

 any of the branches cf the parent varieties : and I feel per- 

 fectly confident that some of the new varieties, and particu- 

 larly one of them, will succeed in forming blossoms, and ri- 

 pening fruit in seasons and situations too cold for either of 

 the old varieties from which they sprang. 



Buds of any of the varieties, which you may think deserv- 

 ing culture, shall be sent in the proper season. Having pre- 

 served and given a place to the original tree upon my wall of 

 one (which I believe you agreed with me in thinking the best, 

 and to which you proposed to give the name of the Downton 

 Nectarine) I shall be able to supply a much larger number of 

 buds of that, than can be wanted. 



I remain, my dear sir, sincerely yours, 

 Downton, Feb. 28, 1824. Thomas Andrew KNIGHT. 



* Note by the Secretary. 



April 17, 1824%— Mr. William Christie, the Fruit and Kit- 

 chen Gardener of the Society, having been at Downton du- 

 ring part of the last week, was much struck with the appear- 

 ance of the blossoms of the new nectarines mentioned by the 

 President in the above paper ; they were particularly plump 

 and strong, and their colour very blight and lively, all indicat- 

 ing vigour of constitution in the branches producing them. 



XXXIV. On the complete Solution of certain Functional Equa- 

 tions. By John Herapath, Esq. 



[ ET us take the well-known equation 



tyx + fx. tyctx =f,x (1) 



the conditions being oPx =x, fx.fctx = 1, and fx.f t *x —J)X* 

 Substituting vtyx + u<px for xf/x wherein v= 1, and b = 0, <px 

 being any arbitrary function, we have 



v^x + bfx + j^ =f,x. (2) 



Then changing x into ax, our conditions give 



vfx.^dX + b fx.fax + tyx —fux.Jx —J\x (3) 



and 



