58 



Transactions of the Royal Canadian Institute 



(14) 



accounts for the presence of three formulae in the second sub-system 

 of (9). 



There is a precisely similar explanation for the presence of four 

 formulae in the third sub-system of (9). 



The numerator for the reduced fraction built up from the five com- 

 ponents of (13) is 



v-|-5.v-f6.v+7 (v+O)"' 

 -4.i; + 2.v+6.»'+7 (i'+2)'" 

 +6.i' + l.i^+4.j'+7 (H-4)'" 

 -4.1"^ l.i'+2.v+6 (i'+6)'"' 

 -f v + l.v + 2.v-^2, (v-fS)'" 



and the denominator is j'-|-1.j' + 2 .... v-\-7. 



The apparent order of the numerator is w-f 3. The requirement for a 

 zero value of the reduced fraction embodied in the inequality m+3 <7 

 furnishes only the values w = 0, 2, whereas w = 4, 6 are known to be 

 admissible from the third sub-system of (9). The explanation is that the 

 polynomial (14) contains no terms in v*"+^ v""^^ , v^^^ , v^\ its true 

 order is w — 1 and the inequality w-|-3 <7 has to be replaced by w — 1 < 7 

 thus admitting the further values w = 4, m = 6. In this way we have 

 accounted for the four formulae in the third sub-system of (9). 



The manner in which it comes about that the terms in v^^^ , p'"'^^ , 

 ^m+i ^ ^m (disappear can possibly be best appreciated by the following 

 analysis: 



Expanding the terms in the right-hand column of (14) we get 



(sum of the left-hand column) +mj/'"~^ -4.i' + 2.i/-f6.j'-f7.2 



+6.j'+l.i'-|-4.v+7.4 

 -4.i^+l.v+2.i'+6.6 



-\- v + l.v-{'2.p-\-3.S 



m{m-\) ^_2 

 + r-: V 



-4.j'+2.v+6.j'-f7.22 

 -1-6.1' -+-l.i'+4.i^+7.42 



-f v + l.v-\-2.v-\-2>.'&'' 



tn(m-l)(m-2) ^_3 

 "^ 3! 



-4.v-^2.v-{-6.p-\-7.2^ 

 ^Q.v-\-l.v-\-4:.v+7A^ 

 -4.v-]-l.v-\-2.v-\-6.Q^ 

 -\- p-{-l.v-\-2.v+3.8^ 



-t- terms in v of orders lower than the m^\ 



Now the coefficient of p"" in the first term is zero; the coefficient of 

 wj*"*"^ is without the terms ^'^ p"^, p (and has the value —240); the 



