160 PROFESSOR KELLAND, ON THE 



Equating coefficients, we obtain 

 «i = 1, 

 «2 + 2«, = 

 th + 2« 2 + 3«i = 

 « 4 + 2as + 3«2 + 2a, = 

 a 5 + 2« 4 + 3«s + 2«2 + a, = 

 a 6 + 2« 5 + 3a 4 + 2a 3 + a 2 = 



«*,+, + 2a 3m + 3a3 m _, + 2(h m _ 2 + flfc._» = - 2, 



= 0; 



,*. a x = 1, a 2 = — _2 

 «3 = 1, a 4 = 2, a 5 = — 4 

 a 6 = 2, a 7 = 3, a 8 = — 6 



a, = 3, a, = 4, a u = - 8 



Suppose this law to hold true for any three consecutive terms ; 



that is, let <h n = »» 



chn + 1 = n + 1, 



«3 B + 2 = - 2«- 2;J 



• '. «3»+3 = 4w + 4 — Sn — 3 - 2w + 2« 

 = ft + 1, 

 a3„+4 = — 2n — 2 + 6n + 6 — 2w — 2 — » 



= n + 2, 



03„ +5 = — 2» — 4-3» — 8 + 4» + 4-«-l 

 = — 2« - 4 : 



hence the law holds good for the next greater consecutive terms, and is 

 consequently general. 



