52 Prof. Sylvester on the Multiplication of 



Similarly, 



C*B*A*= fe^Hf +h+&>v+vi+v 2 n* 2 , y,m+&,v+vi)' 



= A 2 + A o 2 C 3)2e A ^B 21 A 10 ^ 



= e A 12 +A 02 +A 0I C 32 B 21 A 10 , and so on. 



This seems the easiest proof of your general theorem." 



The reader of sufficient intelligence to understand the theorem 

 itself will have no difficulty in supplying the few missing links 

 between my statement and the above demonstration of it. I 

 will content myself with appending a single example to illustrate 

 its meaning and mode of application. 



Let P be any lineo-linear functions of x, y, z, . . ; 8 X , S y , 8 e , . . . ; 

 and in general let 



(P*)»-ip = P n . 



Let it be proposed to expand 



P**P>*. 

 Here calling 



F=r<fc, B> = <£ 2 , 



Ai, 2 = 8 *, i $*, 2 + S' y , I&y,2+ ... 



It is easily seen* that 



A 1(1 (^)=:i.iF-'.p--'.P*P 



(A 1 , 2 )^(^ 1 ^)=i(i-i)P i - 2 .i(i-i)P-'- 2 , 



Hence 



F*Pf* = (P. . Pj)*y F+>- 2 . P 2 * + ''^'"^'^W -'Pg 8 * + &c. 



agreeable to the theorem given for protractors, and stated subse- 

 quently to hold good for pertractors in the previous paper, P 2 

 here denoting what was called 2P 2 in the passages referred to. 



* Thus, ex. gr., to fix the ideas, observe that 



2(8'*, i . bx^axby+byb^cyhe+dzbt) 



—ax . cb z +by . dbt={axb y +byb z )*{cyb z +dzb t ). 

 So again, 



(*'* i • *v, «) a WCy*#y=*(»- 1 ) ju- i) • W- a W-». * 2 (s*) 2 



= i(i-l)j{j-l)(xdr,y-*(y8 z y-z(xdy*yd g y. 



