153 Rev. C. L. Dodgson on Condensation of Determinants. [May 17, 



block (reckoning from the corresponding corner), and transpose rows or 

 columns cyclically until it is left in an outside row or column. When the 

 necessary alterations have been made in the derived blocks, it will be found 

 that the cipher now occurs in an outside row or column, and therefore 

 need no longer be used as a divisor. 



The advantage of cyclical transposition is, that most of the terms in the 

 new blocks will have been computed already, and need only be copied ; in 

 no case will it be necessary to compute more than one new row or column 

 for each block of the series. 



In the following instance it will be seen that in the first series of blocks 

 a cipher occurs in the interior of the third. We therefore abandon the 

 process at that point and begin again, rearranging the given block by 

 transferring the top row to the bottom ; and the cipher, when it occurs, is 

 now found in an exterior row. It will be observed that in each block of 

 the new series, there is only one new row to be computed ; the other rows 

 are simply copied from the work already done. 



2 

 1 



— 2 



— 1 



-2 — 1 

 5 -5 

 3 -3 

 3 3 

 5 -3 

 -30 

 



6 - 



1 - 



— 1 



— 1 - 

 -2 - 

 -1 



-3 -1 

 -3 3 

 3 -1 

 -1 -5 

 6 —12 

 6 

 •6 8 



2 



— 1 

 1 



-2 

 -1 



-3 

 3 



-5 

 3 



-3 



3 3 



—3 — 1 



-5 I 



-1 

 -1 

 -2 

 — 1 

 1 



3 



2 

 -I 

 -1 



2 



-3 

 3 

 -1 

 -5 

 1 



6 

 6-6 8 

 -17 8 -4 

 I 12 

 I 18 40 

 36. 



The fact that, whenever ciphers occur in the interior of a derived block, 

 it is necessary to recommence the operation, may be thought a great 

 obstacle to the use of this method ; but I believe it will be found in prac- 

 tice that, even though this should occur several times in the course of one 

 operation, the whole amount of labour will still be much less than that in- 

 volved in the old process of computation. 



I now proceed to give a proof of the validity of this process, deduced 

 from a well-known theorem in determinants ; and in doing so, I shall use 

 the word "adjugate" in the following sense : — if there be a square block, 

 and if a new block be formed, such that each of its terms is the determi- 

 nant of the complemental minor of the corresponding term of the first 

 block, the second block is said to be adjugate to the first. 



