92 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 



the law of formation of the linear functions of the numbers 



C(...) 

 being similar to that which occurs in 



M(a)(6)(c)...}, 



with the exception that the signs are alternately positive and negative, depending 

 upon the numbers of integers in the brackets. 



Art. 38. Denote this linear function of the numbers C(...) by 



fc{ 

 8othat N (ate...) = 



When it is necessary to put in evidence the numbers whose permutations are 

 under examination we may write the two formulae 



e y {(a)(b)(c)...} {pqr .., = C(abc...) (pqr ..,; 



SECTION 4. 

 Digression on the Forms y , </> c . 



Art. 39. Define in general, so that 



N {(<*!. . .a.^a.) (&!&.,.. . &_ A) (ciC 2 . . .c m -iC m }^d t . . .d e ). . .(A^. . .k,)}, 



where there are k symbols a, b, c, d, ..., k, denotes the 2*" 1 terms forming the series 



N (<*!...*,) 



.&,_i, &+Ci, c a ...k z ) 

 + ... 



+ N (a,. ..a,-i, a.+bt, b 2 ...b t - l , 

 + ..., 

 where additions take place, 



0, 1, 2, ..., k I at a time between the pairs a,, b^ ; b t , Cj ; C B , c/, ; 

 Art. 40. Similarly define 



< c {(<*i.. .,_!<*,) (&i6 3 . . .6-A)(c 1 Ci...C^ 1 c.)(d 1 d f . . .d v ). . .(k^. ..k,)} 

 to denote the 2*" 1 terms forming the series 



!...,_!, a.+bi, &,...*,) 

 C (a,.. .&,_!, 6, + Cx, c,...k.) 



a 1 ...a,_ 1 , '. + &,, b a ...b t -i, 



