1866.] Rev. C. L. Dodgson on Condensation of Determinants. 151 



terms. These values will constitute a second block, consisting of n—l 

 rows and 1 columns. 



(3) Condense this second block in the same manner, dividing each term, 

 when found, by the corresponding term in the interior of the first block. 



(4) Repeat this process as often as may be necessary (observing that in 

 condensing any block of the series, the Kh for example, the terms so found 

 must be divided by the corresponding terms in the interior of the r — 1th. 

 block), until the block is condensed to a single term, which will be the 

 required value. 



As an instance of the foregoing rules, let us take the block 







1 



-1 



-4 







-1 — 



2 



— 1 



-6 







— 1 — 



1 



2 



4 







2 



1 



-3 



-8 









3 



-1 



2 



condensed into 





-1 - 



-5 



8 







1 



I - 



-4 



rule (3), is condensed into 



this, again, by 



1 1-4 1 



^ ~2 ; and this, by rule (4), into —8, 

 — 4 6 



which is the required value. 



The simplest method of working this rule appears to be to arrange the 



series of blocks one under another, as here exhibited ; it will then be found 



very easy to pick out the divisors required in rules (3) and (4) . 



-2 



-1 



— I 



2 



I 



— 1 



-4 



2 



-1 



-6 



■1 



2 



4 



1 



-3 



-8 





-1 



2 





-5 



8 





1 



-4 



8 



-2 





4 



6 



-8. 





3 

 -1 

 1 



This process cannot be continued when ciphers occur in the interior of 

 any one of the blocks, since infinite values would be introduced by em- 

 ploying them as divisors. When they occur in the given block itself, it 

 may be rearranged as has been already mentioned ; but this cannot be done 

 when they occur in any one of the derived blocks ; in such a case the 

 given block must be rearranged as circumstances require, and the operation 

 commenced anew. 



The best way of doing this is as follows : — 



Suppose a cipher to occur in the hth. row and kih. column of one of the 

 derived blocks (reckoning both row and column from the nearest corner 

 of the block) ; find the term in the hth. row and /dh column of the given 



