You can check:

Δαφέρμος, Β. (2005). Κοινωνική Στατιστική με το SPSS. Θεσ/νίκη: Εκδóσεις ΖΗΤΗ. (p. 594) [I think that there is a second edition].

or

Bluman, A. G. (2011) Elementary Statistics - A Step by Step Approach (8th edition). Boston: McGraw-Hill. (p. 642)

or

Gravetter, F. J., & Wallnau, L. B. (2011). Essentials of statistics for the behavioral sciences (7th Edition). CA: Wadsworth Publishing Company. (p. 770)

Thank you Menelae ! I have just checked the books that you proposed and I really appreciate your help!

There is the Scheffe formula in these books but I would also like to know the mathematical proof of this test. Μy fellow students try to proof this statistic with the maximum likelihood method but we think it is a hard work..

I am not sure if this is what you are looking for, but you can check this out:

http://allianthawk.org/victionary/showdef.php?word=289

You can check those two references out for that:

1) Scott E. Maxwell and Harold D. Delaney (2004). Designing Experiments and Analyzing Data: A Model Comparison. Lawrence Erlbaum Associates, ISBN 0-8058-3718-3, pp. 217–218.

2) George A. Milliken and Dallas E. Johnson (1993). Analysis of Messy Data. CRC Press, ISBN 0-412-99081-4, pp. 35–36.

Thank you very much Professor Okasha!

You're welcome!

The best source to find an answer to this question is of course Scheffe's book "The Analysis of Variance", Wiley 1959.

Scheffe's proof is geometrical using parallel tangent planes. An algebraic proof is much shorter (e.g., Seber and Lee, Linear Regression Analysis, 2nd edit. page 123, also in the first edit.