508 TRAXSACTIONS OF SECTION A. 



representation of the function in any poi'tion of the interval in which that 

 function is continuous. It was shown that the subsidiary functions employed by 

 Hamilton are a particular case of a more general class of functions, the pro- 

 perties of which have been recently discussed by the author in the ' Proceedings 

 of the London Mathematical Societv,' vol. vi. 



4. On the Laiv of Eqnipartition of Energy between Correlated Variables. 



By S. H. BuRBURT, F.R.S. 



I think the law has never yet been proved except on the assumption, express 

 or implied, that the variables are not correlated. 



Let, then, .Tj . . ,7„, be n quantities which vary continuously between limits 

 given by 2OT.r- = E, a constant, and which are, or some of them are, correlated 

 inter se. Let the chance that, su!)ject to constants, they shall respectively lie 

 between .r, and .r, +rZ.t„ .r, and .r, +</.?•,,, &c., be denoted by 0(.r, . . .Vn)d.ri . . dx,,. 

 Then their mean values, .r^, .r7, &c., are 



••'"a "^ jj • ■ l^i'^i • • ■'''n)-'<''id.r^ . . d.r„, See. 



Also their mean squares are 



•rj- = JJ ' • <^(<'i • • ■r„).)Vf;.r, .- , dx„, &c. 



And by virtue of the correlation any two of them, as Xj, and ,r„ have a mean 

 product 



■''>'''i -JJ • • 4>(-h . ■ x„\r,,x,,dx, . , <•?.(•„. 



For my present_purpose I must assume that every x = 0. Also let x^=a„ 



.r./ = «,, &c., and .i-jr, = *,o = 60, .... .r^=6;,, = 6„„ &c. 

 Introduce now n new variables 7/^ . . i/„, such that 



.r, = i6._,, /<,+«..?/,, + .... I- .... (1) 



And by consequence 



«,=-r,^W.r,^^+.V3^' + . +-r«%^ .... (2) 



with corresponding values for u^ . . w„. 



Here D is the determinant of the coefficients a, b as they stand in (1), and 

 D,„, is its minor, omitting row^ and column q, or vice versa. 



The laiv of probability is then 



•^(.r^ . . .r,!)=e- !('•.«. + >■>».+ • +^„«„) 



"Itiplied by a constant. 

 In the first equation of (1) multiply each side by .r,. That gives 



x,u, = .r,' ^IL + .r,.r, ^^- + .r,.r, ^^. + &c. , . . . (3) 



Now the values of the coefficients a, b, and therefore of D and its minors,, do 

 not change between one set of values of x^ . x„ and another, but are the same 



