696 DR ALEXANDER CRUM BROWN ON AN 



molecules on which cp can act. Thus, if </> be the replacement of H by COHO, 

 the monocarbon acids, the dicarbon acids, and the tricarbon acids, have the 

 formulas cp X, 2], X, $ 3] 'X, respectively. 



I shall only consider further the application of this method to two other kinds 

 of series, which we may call functional chemical series. The first consists of terms 

 of the form a, (pa, cp 2], a, .... <p" h a. Here the successive terms are produced 

 by the repetition of the process <p independently on a, and we may call such series 

 independent functional series. We have instances of this in the series CH 4 , 

 CH3CI, CH 2 C1 2 , CHCI3, CC1 4 ; NH 4 I, NH 3 CH 3 I NH 2 (CH 3 ) 2 I, NH(CH 3 ) 3 I, N(CH 3 ) 4 I; 

 C 2 H 2 , C 2 H 4 , C 2 H 6 , or in the so-called homologues of benzol C 6 H 6 , C 6 H 5 CH 3 . 

 C 6 H 4 (CH 3 ) 2 , &c. In all these series we have a common difference between 

 successive terms, and we may observe that all such series necessarily consist of 

 a definite number of terms, for the process (p can only be performed independently 

 on a a definite number of times. If we put a — ep-~" ] -b, where b is not of the 

 form cp~ 1 -x,n is the number of times <p can be performed independently on a, 

 and n + 1 is the number of terms in the series. 



The other form of functional series is «, cp-a, <p 2 -a. <p"-a. In this the 



successive terms are derived by the repetition of </> on that part of the molecule 

 which was introduced or modified by the previous performance of <p, and we may 

 call such a series a successive functional series. In order that such a series may 

 be possible, it is necessary that a be of the form (p~ 1- b, and that cp be of the 

 form (p~ 1 -^\ in other words, that cp can be applied once to a and once to (p. We 

 have examples of this kind of series in "homologous" series, such as NHJ, NH 3 

 (CH 3 ^I,NH 3 (CH 2 CH7)I,&c. 



Here the law of derivation of successive terms does not, as in the independent 

 functional series, contain in itself a determination of the number of terms of 

 which the series consists, except where (p diminishes the weight of the molecule 

 on which it acts. In this case, however, hy inverting the series, we find that 

 there are an indefinite number of terms before a, so that by inversion, if neces- 

 sary, all such series may be reduced to the form a, cp~a cp"-a, where a is 



not of the form <p'%, and is, therefore, the first term. 



In a series of this form, if <p does not involve x in the expression cp x, we 

 have a common difference in weight and composition. This is the case in 

 "homologous" series. If cp involve x, there cannot, of course, be a common 

 difference ; this case will, however, be examined in a subsequent part of this 

 paper. 



There are two varieties of the successive functional series — 1st, Where cp is a 

 simple addition ; and, 2d, Where cp is a replacement. In the first, the radical 

 (or radicals) necessarily artiad, must be such that it (or they) can be again added 

 to that part of the molecule which has been introduced or modified by the 

 first cp. We have an example of this kind of series in the polymers of acetylene, 



