4 14 Mr. Ha worth's ThirteenthDecade of New Succulent Plants. 

 log (1 + Aai) + log (1 + Xa.2) + log (1 + \a„) = coeffi- 

 cient of — in . log -\J-. This coefficient is a func- 



X \+Xx ° x'" 



tion of X, suppose \^(A). Hence (l + Aaj). (l—^a.^) 



(1 +Xa,„) = e'^'^^^ s being the base of Napier's logarithms. 



Let e '^^^^ be expressed in powers of X, the general term 

 being Up . AP, and suppose we equate the terms involving like 

 powers of A, on both sides of the equation ; then 



First ; When j5< m, iip is evidently the sum of the products 

 of the m least roots, taken p and p together. 



Secondly ; When p = m, Up = the product of the m least 

 roots. 



Thirdly ; When p '>ra, tip is zero, 



From the third case, it follows, that in applying this method 

 we may always reject from ^'(A) all the terms after that which 

 involves X". 



A process exactly similar will give the combinations p and 

 p together, of any functions of the m least roots or their con- 

 tinued product. I am, Gentlemen, your obedient Servant, 



R. Murphy. 



LIV. Decas tridecima Novarum Plantarum Succulentarum ; 

 Autore A. H. Haavorth, Soc. Linn. Lond. — Soc. Horticult. 

 Lond. — Soc. Cccs. Nat. Curios. Mosc. — necnon Soc. Reg. Hor- 

 ticult. Belgic. Socius: 8)C. Sfc. 



To the Editors of the Philosophical Magazine and Annals. 

 Gentlemen, 



AFTER a longer space than usual between my communi- 

 cations to your valuable Magazine and Annals, I send you 

 hereunder my thirteenth Decade of New Succulent Plants, 

 for insertion, if you please, in an early Number of that useful 

 work. 



As heretofore, I have carefully stated the native countries of 

 every species, — detailed, but with due precision, their botani- 

 cal characters ; and not failed to record the rich gardens from 

 whence I have received them. And to all these particulars 

 I have added such contrasting and other characters as may 

 enable the gardener to grow them, the botanical tyro to un- 

 derstand them, and every able author to appretiate and blend 

 them with their old affinities in a scientific way. 

 I remain, Gentlemen, yours, &c. 

 Chelsea, Aug. 30th, 1831. A. H. Ha WORTH. 



Ord. 



