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[34 
34. 
NOTE ON THE MAXIMA AND MINIMA OF FUNCTIONS OF 
THREE VARIABLES. 
[From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 74, 75.] 
If A, B, C, F, G, H, be any real quantities, such that 
BC + CA + AB - F 2 — G 2 - H 2 , 
and (A+B + C) (ABC - AF 2 - BG 2 - CH 2 + 2FGH) 
are positive; the six quantities 
BC - F 2 , CA - G\ AB - IF, AK, BK, CK, 
(where K = ABC — AF' 2 — BG 2 — CH 2 + 2FGH) are all of them positive. It is unnecessary 
to point out the connection of this property with the theory of maxima and minima. 
To demonstrate this, writing as usual 
BC-F* = A', GH - AF= F', 
CA -G 2 =B', HF - BG = G', 
AB~H 2 = C', FG - CH = H\ 
and K as above: then if A", B", C", F", G", H", K' be formed from A', B\ C', F', G', H', 
as these and K are from A, B, C, F, G, H, we have the well-known formulae 
A" = KA, F" = KF, K' — K 2 . 
B" = KB, G" = KG , 
C" =KC, H" = KH,