352 ' MEMOIRS OF THE AMERICAN ACADEMY. 





We should thus readily find 



(120.) d*{&) «[*].[•— i].[»— 2]. ...[» — m+ i^.x n ~^,{dx) 

 Let us next differentiate l x . We have, in the first place, 



% 



Al x — /x -^ Ax _ l% — p px — fr=z l*,(l 



Ax 



The value of P x — 1 is next to be found. 



We have by (111) 6^*-i — /ax^ 



Hence, 



/Ax — 1 — log^A*.. 



But by (10) log/** = (log J) As . 



Substituting this value of id* — 1 in the equation lately found for dl x we have 



21.) dl x = l x ,{\og,l) dx = l x ,{l — l)dx = — /*,(! —T)dx. 



In printing this paper, I here make an addition which supplies an omission in the 



account given above of involution in this algebra. We have seen that every term 



which does not vanish is conceivable as logically divisible into individual terms. 



Thus we may write 



s = S' -fc- S" -t- S'" + etc. 



where not more than one individual is in any one of these relations to the same indi- 

 vidual, although there is nothing to prevent the same person from being so related to 

 many individuals. Thus, " bishop of the see of " may be divided into first bishop, 

 second bishop, etc., and only one person can be tT th bishop of any one see, although the 

 same person may (where translation is permitted) be tr th bishop of several sees. Now 



let us denote the converse of x by JfiCx ; thus, if s is " servant of," JfCs is " master or 

 mistress of." Then we have 



JCs = JCS' -fc JCS" -t- 3CS'" -fc- etc. ; 



\ 









and here each of the terms of the second member evidently expresses such a relation 

 that the same person cannot be so related to more than one, although more than one 

 may be so related to the same. Thus, the converse of " bishop of the see of — " is 

 " see one of whose bishops is — ," the converse of u first bishop of — " is " see whose 

 first bishop is — ," etc. Now, the same see cannot be a see whose tl th bishop is 

 more than one individual, although several sees may be so related to the same indi- ^ 





