MA Jon P. A. MAC-MAHGN ON THE COMPOSITIONS OF NTMI:I i> 129 



Art. 111. We have seen that, in general, we have two expressions for 



but, since ^ _ jj 



we have four expressions for -^ 



* M P* 



viz " /N w +- 



P 

 ~ . (N r/ , 



( '*P +P )* 



(V)' 



Art. 112. It is clear that the operator relations afford unlimited scope for obtaining 



theorems connecting the numbers 



N... -,,, 



Relations, so far utilised, have involved the operator 



D,, 



but it is easy to construct them so as not to contain D, and generally so as not to 

 contain D. , where $ is less than a given integer. 



E.g., from the symmetric function relation 



we find 



D,' = & S D,-6&D 3 +6D 4 ; 

 and generally the relation 



leads to 



or, throwing out the factor 



D,' = fc1),-6&D 3 +6D 4 , 

 the same relation as before. 

 Moreover, the relation 



(pq) (rs) = (p + r, q + s) + (p 



+ ( p + *, qr) + (q + s, pr) + (pqrs) 



leads, after throwing out a power of b t to precisely the same relation. 



Art. 113. This remarkable circumstance greatly limits the number of operator 

 relations obtainable. It should be observed that any operator relation may be 

 multiplied throughout by any power of b and may be then used to obtain relations 

 between the numbers XT 



VOL. CCVII. A. 



