54 Transactions of the Royal Canadian Institute 



The numerators constitute the series v, v-\-2, v-\-4, v-\-Q, . . . ., the 

 numerical coefficients are the binomial coefficients with signs alternately 

 plus and minus, and the denominators follow a law that is evident on 

 inspection. 



The complete system (6) can be written 



v.v + l.v-\-2 v + l.v + 2.j'+3 v + 2.v-\-S.v-{-4: 



v.v-{-l.v+2.v^-Z J/ + 1.Z/ + 2.V+3.V+4 i' + 2.v+3.j'+4.j/ + 5 



(»'+6)^ 



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







_4 (^+2)^ +6 ^''+^^' 



(8) 



j'.i'+l.y+2.i/+3.j'+4 v^-l.v-\-2 . . . v-\-b J/+2 . . . J/+6 



j'+3 . . . 1^+7 v+4 . . . v+S 



The resemblance in structure between (7) and (8) is at once apparent. 

 The denominators are the same in corresponding terms; there is the 

 same law as to the numerical coefficients and the only novelty consists in 

 replacing the first powers of v, v + 2, ^+4, v+6, ... by third, fifth, 

 seventh powers, .... 



The second equation of (7) and the second equation of (8), containing 

 as they do in their numerators the first and fifth powers of v, v-\-2, j'+4, 

 v-\-Q, suggest an intermediate equation 



_3 (v+2)^ _^3 {^+4:y 



v.v-^l.v-]-2.v-\-S j; + l.v+2.v+3.i'4-4 ^ + 2.v+3.i'+4.i'+5 



(^+4)« =0. 



v-\-S.v-\-4:.v-\-5.v-\-Q 

 Similarly the third equations of (7), (8) suggest intermediate equations 

 p' _4 (^-+3)^ _^g (»'+4)^ 



v.v-\-l . . . v-\-4: v-\-l . . . v-\-5 v-^2 . . . v+Q 



v+S . . . v + 7 v+4.. .1^+8 



-4 ('-+2)^ +6 (''+4)^ 



v.v-[-l . . . v-\-4: v-\-l . . . v+5 v-\-2 . . . J^+6 



v+3 . . . v-\-l v+4 . . . »'+8 



