On a peculiar Determinant of the Sixth Order. 429 



In conclusion, it need hardly be pointed out that, even if 

 these criticisms be admitted, they do not impugn the great 

 accuracy of Mr. Pickering's results, but only the extreme 

 accuracy necessary for his purpose ; further they do not of 

 course disprove the hydrate theory, but only reduce it, so far 

 as they go, to the condition known in Scottish law as not 

 proven. 



LI. Note on a peculiar Determinant of the Sixth Order. 

 By Thomas Muir, LL.D* 



IN the Philosophical Magazine for March 1856 (vol. xi 

 pp. 378, 379) Professor Cayley gave, as a bye-result o 

 a process of elimination, the curious identity 



. C B -2A' 



C . A . -2B' . 



B A . . . -2C 



A' . . A -C -B' 



. B' . -C B -A! 



. . (y -B' -A' C 



of 





A 



C 



B' 



C 



B 



A' 



B / 



A' 



C 



or, as he shortly writes it, 



D = -2K 2 7 



concluding his note with the remark, " It would be interesting 

 to show a priori that D contains K 2 as a factor." The 

 appositeness of the remark is not a little enhanced when it is 

 recalled that Sylvester, who was the first f to light upon D, 

 made it out to be equal to 



ABC(ABC-AB /2 -BC /2 -CA /2 + 2A / B / C / ), 



i. e. 



ABC 



A A' C 

 A' B B' 

 C B / C 



The object of the present short note is to supply Professor 

 Oayley's desideratum. 



* Cornmimicated by the Author. 



+ See his ingenious paper, "Examples of the Dialytic Method of 

 Elimination as applied to Ternary Systems of Equations," Cambridge 

 Math. Joura. ii. (1841) pp. 232-2315. 



