

Differences of the Second Order 



between them. 



thus obtaining 









h 



a 3 



a 2 



a ly 





a 5 



h 



a 2 



<h> 





a 5 



a± 



b 2 



a x , 





a 5 



a 4 



a 3 



h; 



117 



in every possible way leave out of the first term two such pairs 

 and insert instead the two b's corresponding, thus obtaining 



& 4 b 2 a ly 



b± a B b ly 



7 % h hi 



ana so on. 



The coefficient of b u manifestly follows the same law. 



3. The importance of this rule of Professor Cayley's is not 

 to be measured by viewing it in the special connexion in which 

 it is found enunciated. It is a general rule for the evaluation 

 of continuants, and is sure to be of good service wherever these 

 functions make their appearance. 



4. When no insertions follow the omissions referred to in the 

 rule, we obtain the development of a simple continuant — in the 

 above example we should find the development of K(a 5 a 4 a 3 a 2 a 1 ). 

 It must be noted, however, that the rule as thus reduced has 

 been long known, probably since the time of Euler. 



5. But now suppose that the given equation of differences 

 is in the more general form 



Ux == a x — \U X ^. i T ##— 2^x— 2 i £#— 2; 



and we employ the same method as above. It is easy to see 

 that, on solving this time for u 6 , we shall obtain 



w 6 = 



a 5 6 4 c 4 









— la463.Cs 







. — 1 a B b 2 c 2 



m 





. —1 a 2 biUi + Ci 



9 





. —1 aiUx + boUo + Co 





and that thence we shall find 



K / h h b 2 h \ +bK ( h b 3 i, 



b \a 5 a± a B a 2 aj L u \a 5 a± a z 



+ 



a 5 64 . • c 4 



— 1 a 4 b 3 . c d 



. — 1 a 3 b 2 c 2 



. — 1 a 2 c x 



• 







. -1 c 







)u 



