TRANSACTIONS OF SECTION A. 599 



for all values of .rj . . .r„ and the consequent values of w, . . w,j. Therefore 

 we may multiply both sides of (3) by ^(.tj . . .%„) and integrate for all values of 

 .ij . . Xn, regarding a, b as constants. And since 



we obtain 



A' 



l"l = J7^ • ■ "^^-''l ■ • •'■"^'^•■''l • • '^"'"{•'l'^ + ''r'. ^ + &C.| 



D 



Similarly 



.r^Mj = 1, &c. And .t\Ui — x.m^ = &c. 



Or if 



Q = |2.rM, ai4^ = x,^ = &c. 



This is the law of equipartition of Q. 



I now introduce the following principle : — 



Making Q minimum, i.e., e~^ maximum, subject to the kinetic energy E 

 being constant, we find the most probable, or normal, state of the system. That 

 is, since Q is minimum 



■^^ ax 

 and since E is constant 



whence by the usual method 



^ = X — i^ = X. ^^ &c 



dx^ dx^ ' dx.^ dx^ ' 



The indeterminate multiplier X is a function of the a's and b's. Therefore we 

 may write 



dQ . dE dQ . dE 



dXj^ dx^ dx^ dx^ 



And this being true for all values of .rj . . .t„ consistent with E = constant 



dE ^. dE _ dQ dQ _ 



dx^ ] ■ dx,, da. I I ~ dx.^ 



That ' ' ■ __ 



, dE _ . ^E 



.1" -j — T.2 -T — I 



And if E = 2mx^, m^x^ = 7«jao'-, &c. 



5. The Complementary Theorem. By Professor J. C. Fields. 



Let F (2, t;) = be an integral algebraic equation. Any rational function can 

 be written in the forms 



(1) HC, .) = P(. ^ 2 ^^^- = - ^^^^ . i^ ((^, .)), 



