252 On Binomial Biordinals. 



86. Since 



a) 2 + 5a> + 6 = 1 LV = 1 LU 2 -iU, 

 therefore 



0) = JU — 3, or else &>=— |U- 2; 



Also, by art. 79, 



r T 2 — J 2 1 



87. Again, by art. 69, 



y:=e f(p-Wtg = ^ gee artg> 71 and 73^ ^ 2 -a T ( W+ 2) ?= ( see 

 art. 85) ^-a+b T (o>+2) Z = ( see arts# 79j 86 ) ? 

 ^ a -i T ( w +2) Z=: ^ T (co+2) Zj wneri; as i n art . 83, a 2 = l. 



88. The shorter process (see art. 81) is this. By art. 78, 



/(D)=D 3 + 3(a 2 -l)D 2 +{3(m 1 -a 2 )+2[D + m 3 



= (D + a 2 -l) 3 -a(D + a 2 -l)+/3 

 suppose. Also (see art. 77), 

 F(D) = D 3 + 3(« 2 -0-l)D 2 + {Z(m 1 + m 2 -a 2 + 0) + 2}T> + m a + mt 

 = (D + a 2 -l-dy-ry(l> + a 2 -l-0) + -R 

 suppose. Then 



F(D) -/(D) = -30D 2 + 3(™ 2 + 6>)D + m 4 



=: _36>(D + a 2 -l) 2 +(3^ 2 -7 + «j(D + a 2 -l)+H- / e + 76'-^ 3 . 



89. Hence, accents now meaning derived functions, 

 3m 1 =/ / (0)+3a 2 -2 = 3(a 2 -l) 2 + 3a 2 -«-2, 

 3m 2 =F / (0)-/(0)-3^=-6a 2 6> + 3(9 + 36' 2 -7 + «...,(22) 



W4 =F(0)-/(0) = -3(9(a 2 -l) 2 + a 2 (36' 2 -7 + a) 



-^ 3 -36' 2 + r -« + H-^ + 7(9, 

 m 4 + 3fe 1 -3%m 2 = 6a 2 2 6>-6'(6>-t-l)((9 + 2) + (7-«)(6' + l) + H-/3. (23). 



90. By means of (22) and (23) 7 and H can be found. 

 The results, after all reductions, are 



7 = 3G 2 + - 1 V 2 and H = 2G(a 2 -^J 2 ), 



and the F(D) of art. 82 follows. 



91. When I wrote art. 64 I was not aware of the turn 

 which the discussion would take. The errata indicated at the 

 beginning of vol. ix. of this Series should not be overlooked. 



12 St. Stephen's Road, Bayswater, W., 

 June 3, 1887. 



