Note on Clehsclis Tlieorem. 
397 
and in this the cofactor of \d-,e^f^\ is 
I 64 77ii&i + mj)^ + 77^363 7^^6I + nj)^ + 7^363 
I 77ij.Ci 4- 7?i2C2 + '^'^'3^3 7iiCi + 7i2C2 + 7^303 , 
which 
= T)^m^n2 — D277^I7^3 — D37?i27ii + 'D^m^n-^ + Dzin^n^ — Ti^m^n^i 
Similarly the cof actors of \d^e2f^\, \d^e-^f^\, \d^ej^\ are found to be \'D^m2n^\, 
[D^77a37^4|, |D27?^37^4| ; so that our double-bordered determinant is ex- 
pressible as a sum of products of pairs of three-line determinants. (e) 
9. As this result may be written 
d^ d^ d^ d^ ii 
62 63 64 • 
/x y; /s Ai' 
D, D2 D3 D4 
77ii 77^2 7;z3 
n. 
which again by the multiplication-theorem is equal to 
\a,h,c,d^\ tdm tdn 
we observe that we might have begun by expanding the original determi- 
nant in terms of the minors formed from the first four columns, and 
thence proceeded through {e") and (e') to (e). 
I 
