APPLICATION OF MATHEMATICS TO CHEMISTRY. 69' 



where <p consists in the addition of the radical C 2 H 2 _0_(^)_0 This series 



i i 

 may be represented in several different ways, and we do not as yet know which 



is the true one. I append graphic formulae of two of these ways, to illustrate 



my meaning — 



0@ © © © © (=) © 



GK r©~©^ r? - ? - ?^ r©-©-©-© 

 00 © © © ©A©© 



a (p'a (p 2 'a <p 3 -a 



0(h) ®(h)(h)© ©(*)©(*)©© ® <^) (=) <£> (=) (=) ® © 



©=© ©=©-0-(s) ©=©-©=©-©-© ©=©-©-©-©=©-©=© 



\^_J K J V. J V. y 



The second of these is the form proposed by KekulI:, and appears on the whole 

 to be the most probable. 



In the other variety of successive functional series, <p is the replacement of 

 one or more radicals by one or more new radicals. Let the replaced radical or 

 group of radicals, be represented by I\ and the replacing radical or group of 

 radicals by A, and we at once see, first, that T and A are equivalent; and, second, 

 that A must contain I\ as the process is capable of repetition. When in <p ■ x is 

 independent of x 1 the difference between A and r is constant (as indeed A and 

 T are themselves constant throughout the series), and is the common difference 

 of the series. This is the case in " homologous" series. As an example, we may 



r® 

 take the series of the fatty acids. The formula of one of them is ©^©^ 



©-© 

 where R is of the form C„H 2re+1 ; and taking any of the series of substances 

 through which it is connected with the next higher member of the series, such as 



r© ? ? ® rt <? r® 



©^y ®-dy ®-6-© ®-6-© ©-©-©3© ©-©-(^ 

 ©-© © ob ® © © © ©-© 



we see that r is 0" and (HO)', and A is H', H', and (COHO)'. A thus contains r, 

 and the process is capable of repetition. The common difference is here CH 2 , but 

 there are instances of series probably of the kind we are now considering 

 (although we cannot at present trace the relation between successive terms) in 

 which we have other common differences. Thus the carbonates, oxalates, and 



