Action of Alkalies on Wax. 17 



without any one of the set /3, y, &c. being nothing. The sum 



of the coefficients is half the coefficient of # "*" in V ( I + 4 a?), 

 as is easily proved. 



The readiest way of forming P. , . from P. is as follows: — 



(1.) Put on the next letter to every term of P., and also repeat 



the last letter of each term once more than it occurs already ; 

 thus fli 2 c gives ah* c d + ab* e*. (2.) Put all the results of 

 (1.) together, and correct or introduce coefficients by the law 

 above ascertained. 



It is also worth notice that the portions of P. which do not 



contain any letters beyond a given one foliow the law of a re- 

 curring series. Thus, if P w . signify all that portion of P in 



which nothing beyond k occurs, we have 



P „,* = 4P »-M' P „, C =( A +<)P„-1 1<; . 



and so on ; the coefficients (b + c + d), &c. being derived 

 from the denominators of the ordinary approximations. 



The mode thus given of turning a continued fraction into a 

 series is not so easy in practice as the one derived from invert- 

 ing the process of turning the ratio of two infinite series into 

 a continued fraction ; but the law is worth consideration, the 

 more especially as, from the presence of none but consecutive 

 terms, it cannot be directly connected with Arbogast's methods. 

 This is the reason why the reader looks in vain for anything 

 about continued fractions in the Calcul des Derivations. The 

 most complete account I know of continued fractions is in 

 Eytelwein's Grundle/iren, &c, in which, however, the inverse 

 method occupies only six pages, the direct one eighty-nine ; 

 and no general law is given for the series considered in this 

 paper. 



III. On the Action of Alkalies on Wax. By Robert 

 Warington and W. Francis, Esq.* 



FEW subjects have of late engaged so much the attention 

 -*- of chemists as that relative to the formation of fat in the 

 animal organization, a subject fraught with results of the high- 

 est importance both to science and its applications to rural 

 ceconomy. 



Two theories have been proposed to account for its origin : — 



* Communicated by the Chemical Society; having been read May 16 

 1843. 



Phil. Mag. S. 3. Vol. 24. No. 156. Jan. 1844. C 



