42 THE ROYAL SOCIETY OF CANADA 



[0HW7-J] = x^ |3'i^3'3->'7--3'io \+x^\ yo-y^-y^y^o I + I yib'^-ysysb^io I 

 -\-x^\yi^y7y8^yg^ I + |3'3^3'63'7^3'8^3'^o I + bi^y^ye^yiW I 



+ \yiyiyb^yë^y8^\+x^-\y3yi^yà^y7-\-\- i3'23'3^3'4^3'6^3'"io| 

 + 1 yiy2b'3^yb^y9'^ \+x\ yi^yo^yi^y»- \ 



(xi) (.r^o 1^1 2^2 („. _ !)€« „ 1 ) =, x; I ei 62 .,e„ -1 I _^eo 



the summation to include without repetitions only those addends in 



which ey = eo = w-(ei + e2+ . .+e„.i). 



Examples.- — ^For n = l, €o=l, €i = 2, e2 = l, es = 2, €4 = !, €3 = 0, 

 €6 = 0and .-. r = 0, 2, 4 



{xV- 2 3M) = { |3'i^3'23'3'3'4| + bi'y^y.y^ I } 



= { yi'^^yiyi + y^^y^yiyx + yiy^y'i-y^ + y^-y\yhy-i + y^yzyyi^y^ + 

 :y6b'5:y4^3 + 3'i^3'23'53'6^+>'2^3'43'33'5^ + yiy^yiy^ }x 



For 77 = 7, eo=l, ei = 2, €0 = 2, €3 = !, e4 = 0, €5= 1, e6 = and .'. r = 0, 3, 5 



(xP 2^ 3 5) = { |3'i^3'2^3'33'5| + \yiy\yhy2 \ ]x 



= { 3'1' >'2^3'3 J5 + 3'2"3'4^M63'3 + yz^y^y^yx + 3'4' 3'l">'53'6 + 3'5"3'3-3'l3'4 

 + 3'6^3'5^3'43'2 + 3'3W3'53'2+3'6^3'l^3'33'4 + 3'2^3'5^3'l3'6}.^- 



In the former of these examples |3'33'4^3'536"| is omitted, being the 

 same as |3'i^3'23'3^3'4l, in the latter example |3'5W3'23'4| is omitted, being 



= Wy-fyzyA- 



For n = \\, eo = 2, ei = 2, €2 = 2, e3 = 0, €4 = 2, €5 = 2, €6 = e7 = es = 0, 

 €9=1, €10 = 0, and .-. f = 0, 1, 2, 4, 5 



(x2 P 22 42 52 9) = { |3'r3'2^3'4^3'5'3'9| + |3'i"3'3-3'4-3'83'-iol + l3'2^3'3"3'73'9">'"ioI 

 + l3'i"3'53'7-3'8-3'^]oH- l3'43'6"3'7-3'9"3'"ioI } -^" 



(xii) Kof {iC) = ' the coefficient of 11. ' 



Theorem I. — By the multinomial theorem 



7i(m — l)l , 



'" €i!€2!€3! 6„_l!'^^ ^' '' ^«-1' 



in which w = €i + €2 + e3+ +e„_i . 



Theorem II. — By Waring's theorem 



a h c ^ 



c„,= z{{-ir ^- '^^ ^^ \ 



a\b\ c\ J 



in which T = a-\-h-\-c-\- 



and ar-\-hs-\-ct-\- =m. 



Theorem III. — Hence, in c^ 



Kof |3'i^i3'/^ y^--^\ is 2{(-l)^PiP. Pj 



in which g is to take all possible values. 



