168 Mr. Homer on the [March, 



Article III. 



Oh the Solutions of the Functioii \I/' x, and their Limitations. 

 By Mr. W. G. Horner. 



(To the Editors of the Annals of Philosophy.) 

 GENTLEMEN, Bath, Feb. II, 1826. 



In pursuance of a design long entertained, I request the 

 favour of your inserting in the Annals the annexed investigation 

 of the most useful properties of the formula ■^' x, or <f~'J' <p x, 



yfheaf'x= - — j—. Since the appearance of the Ladies' 



Diary for 1821, containing a perfectly general solution of this 

 equation, I had hoped that my hastily written paper of 1817 

 would not be quoted as the standard of my views on the subject ; 

 but, very possibly, the succinct statement in the Diary may have 

 been overlooked by some of your mathematical readers. They 

 will, therefore, I conceive, be gratified with the present republi- 

 cation of the same theorem, investigated in a rigorous, yet 

 simple and perspicuous manner, and supported by so much of 

 unambitious elucidation as will suffice to convey correct ideas 

 both of the extent and the limitations which are proper to it when 

 applied to ordinal, and particularly to joen'oc^jca/ functions. On 

 the latter subject, Mr. Herapath will perceive that a statement 

 of mine, which he has controverted, is correct, and that an 

 essential distinction exists between '\i- x =. x, or the second 

 order, and those superior to it. The foundation upon which 

 correct theories are to be established, cannot be too cautiously 

 laid. I am, yours, very truly, 



W. G. Horner. 



1. Leaving the ojserations (p and <p~', which are requisite for 

 giving the utmost extension to our conclusions, to be supplied 



by the reader, I proceed directly to the formula f'x=- '—. 



■' ' ^ •' -^ c- + d.x 



Here a., h., &c. represent certain functions of a variable z, whose 

 mutual relations to it and to each other, it is our business to 

 determine, it is a well-known property of this formula, and 

 for the application of which, in the iirst instance, I believe, we 

 are indebted to Mr. Babbage, that, regarded as a function oi x, 

 it always reproduces formulee similar to itself. An important 

 consequence of this remark is, that x can never interfere in the 

 functions of 2;, if it be absent from a, b, c, d, at the casey"a; = 



^ — , assumed as the orieia of the functions ; and, therefore, 



our reasonings, whether they regard y*" a; as a function of «, or of 

 X, cannot clash with each other. 



