108 On Reversion of Series. [Feb. 



The minuteness with which the processes are detailed, scarcely 

 permits us to doubt of their general accuracy, so far as respects 

 the quantity of animal matter which may be procured from bones. 

 Our readers must, however, be aware of the want of correctness 

 in the terms that are employed by the Committee in describing 

 the products which they obtained. The terms " gelee" and 

 " gelatine" are really synonymous; the " gelee" of the first 

 process, when freed from the fat and other extraneous sub- 

 stances, is the only real jelly ; while the "gelatine" of the 

 second process is albumen, a substance different from jelly, 

 which, as Mr. Hatchett has shown, composes the proper animal 

 matter of the bones. 



Article VI. 



On Reversion of Series, as connected with the Equation 4- a ~ l 



4> a. x = x. 



(To Dr. Thomson.) 



SIR, Bath, Nov. 10, 1817. 



The three remarkable cases of reversion of series, alluded 

 to in the paper which you did me the honour to insert in the 

 Annals for this month, deserve to be reconsidered. What I 

 said concerning them was an after thought, too hastily appended 

 to the principal subject ; and on revisal I am sorry to perceive 

 that I have adopted a conclusion which is erroneous. The 

 remarks I now proceed to offer, in lieu of the former, will be 

 found perfectly correct, and some of the results so novel as to 

 atone (I hope) with mathematicians for this noiSo; on^gtmivov. As 

 early an admission as possible will be regarded as a particular 

 favour by Your very obedient servant, 



W. G. Horner. 



Explicitly stated, the cases in question are, 



I. When x = ^ y = a + by + c y 2 + d y s + e y* + &c. 



And y = ^ x = a + b x + c a a + d x 3 + e x 4 + 8cc. 

 This case has been considered by Mr. Babbage in the Journal 

 of Science, vol. ii. ; who has rightly observed, that it is nothing 

 else than a different mode of denoting the general solution of 

 <J/* x — x. 



II. When x = -by = a + by+cy* + dy 3 + e y 4 + &c. 

 And y = — -vj/ — x = — a -\- bx — c x* + d x s — e x 4 + &c. 



III. The singular case, appropriately so called, since only one 

 specimen of it has hitherto been discovered, in which the even 

 powers are absent, and 



x = -\>y — y + a y 3 + b y b + cxf + d y 9 -f e y u + Sec, 



