56 Transactions of the Royal Canadian Institute 



denominator, takes the form 



v+l.i' + 2.j'+3 

 The numerator has the factors 1^4- 1, v-\-2, v-\-3 if m is even, as is seen at 

 once by putting ;'=!, -2, -3. Hence the fraction (10) is only a fraction 

 in appearance, but in reality a polynomial, provided m is even. This 

 gives the answer to (i.)- Furthermore the expression (10), when treated 

 by the method of partial fractions, must have no fractional parts but 

 only an integral part of order w — 2. 



Let us next consider the generalization of the polynomials in the 

 second sub-system, namely: 



— 6 ~ ■ -f-. 



v.v-{-l.v + 2.v-\-3 j^+l.z^+2.i/+3.t'+4 j^ + 2.v+3.v+4.!/+5 



{v + QT^^ 



;'+3.j'+4.v+5.j^+6 



_ v'"{v+^) (i/+5)-3(»^+2)"+H^+5)+3(»^+l) {v+4:r+^-{v^-\){v+2) (v+6)'" 



V+\.V + 2.V^?,.V+4:.V+b (11) 



The numerator, if m is even, has v + l, v-{-2, v+3, v+4, v-\-5 as factors. 



It is desirable to prove this by a method which will apply, mutatis 



mutandis, to the most general case where the expression to be discussed 



is of the form 



j''"+' (u + 2r+' , n(n-l) 



— n -\- 



v.v^l . . . J/+W j' + l.v+2 . . . jz+w + l 2! 



<-+^>"" +... (12) 



j^+2.j'+3 . . . v+n+2 



By the ordinary process associated with partial fractions, we have 

 for the fractional parts (omitting for brevity the integral parts) 



V.V+I.V + 2.V + 3.V + 4. 



(■_1)'« (-1)'"2'" (-1)'"3"' (-1)'"4"* 



1.2.3 + -1.1.2 + -2.-1.1 + -3.-2-1 



j'-l-l v-^2 v+3 j'+4 



(II) _^ (. + 2)"'+^ 



)- (13) 



i/ + l.i' + 2 . . . v + 5 



1 (-1)'" (-1)"'2"' (-1)"'3^ 



-4 I 2.3.4 +*+ -2.1.2 + -3.-1.1 + -4.-2.-1 1 



