54 Prof. Sylvester on the Multiplication of 



quantities of the form 



(P*P)? , (P*P*P)r 



p2/3 p3y 



that can be got consistent with the satisfaction of the equation 

 in integers 2{3 + 3y=r. The upshot of the calculation is that 



- J ffl(/)P»P ., (t)C/Xfc)P»r»p ) 

 P**P'*P** = {e i~v* + p^ f P i +^' +ft }*j 



where it is of course to be understood that [i], (j), (k) are mere 

 umbra, subject to the law above stated for conversion of their 

 powers into factorials of actual quantities. 



The law for any number of factors is now obvious, and may 

 be extended to the case where the factors are powers, not of one 

 single operant P, but of different operants P, Q, R, S, subject to 

 the sole condition of their being linear quantics in regard of 

 $*j S > S z , . . . ; and it will be found thatf 

 F*Q>*R**S'* . . . 

 • -( WP*iM , (0P»(y)Q*(t)R .| (f)p»(j)Q»wR»«)s tM \ -.^j.^^, a 



= {e 2 ^ pq + pqr + pqrs J.P l Q?R*S\ ..}*, 



t Thus, for the particular case when P=Q=R . . . , we have 



p»'i*p«2*p*3 m m m p*«*_ [[eP» ]P2.] * y 

 where 



0=(P+(»i)P*)(P+(i)P#) ... (P+(i.)P#) 

 — P w — 2iP w-1 .P*. 



Observe that if we convene to understand by 

 A* B* C 



the expression 

 then if we write 



L "M "' N ' * 



A*B* . . . *C 



LM...N 5 



the above theorem takes the form 



P -p-- Q =q 



P**Qj'*Hfe* . . . = [e 2 * P» . Q'. R* . . . ]*. 



Now suppose 



A = p» p»' p»" 



*=QJ .Q,..., 



9=R* . . . 



