THOMAS MUIR ON DETACHED THEOREMS ON CIRCTJLANTS. 643 



o 



which, using 2 for cyclic sum, we may write in the form 



la\ afiySe - 2a s be2a 3 yS - ±a 3 cd2a 3 l3e+ hi 2 b 2 dta 2 8y 2 + 2a 2 bc 2 2a 2 t3 2 8 

 — 10 abcdeafiySe . 



Putting a=/3 = y = S = e = l, we obtain 



C(a, b, c, d, e) = 'Ea 5 — 5'Ea 3 bc — 5T,a 3 cd + blla 2 b 2 d + 5'Ea 2 bc 2 \ 



— 10 abcdraftySe , ' 



and it is seen how the coefficients, — 5, — 5, 5, 5 originate. Unfortunately the 

 general determinant still contains a set of unsifted terms, viz., 10 abcdeafiySe, 

 which are got, curiously enough, from the elements by a series of knight's 

 moves. Though therefore the discussion of the problem may be advanced in 

 this way, the full solution is not yet in sight. 



VOL. XXXII. PART III. 5 P 



