622 REPORT— 1895. 



4. To do that, substitute Sj . . . s„ for a^ , . . a„ by the linear equations, and 

 then perform the integration according to b c, &c. That reduces the integral to 

 (f)(s^ . . . s„)ds^ . . . dsn, where (p denotes some function. 



Now let 



Xa= [[ . . ./,€C^.«.^+''.<'A+ ...)v'-ida, . . . da„, 



Xt={{... /^6<M.«.+M,Ma+ ■■■)-^='^ db, . . . dbn, &C. 

 That gives 



XoXb , . . X3= j j . . . 4>{s^ . . . «„)£<«.«■ +«A+ ... V^i ^g^ ^ ^ ^g^^^ 



5. It is now proved (Prop. I.) that 



<^('-i . . . A,)= f t°! . de, . . . rf^„x^x, . . . x,e-"-.«.+ . .. +v«'^^i 



J J —CO 



multiplied by a constant independent of ?•,.., r„. 



6. Proposition II. is next proved, that if 



^du,\a 



\du„ . . ./(?'i . . . u„)cos(6^Uj^ + 6.-,7e.^+ . . .)=pcosr, 

 and 



I du^ <^«2 • • • /("i • • • "") sill (^i"i + ^;jWo + . . . ) = p sin r. 

 And if at the same time 



. . /(Ml . . . u„) = 1, 



\du^ \du^ . 



Then if 5, = (9., = ... =^„=0,p = l. If any ^^0, p<l. 



From this it follows that unless every ^ = 0, each of the quantities Xa,X(, ... X^ is 

 less than unity. 



7. Now let a, . . . «„ be any set of correlated variables, and letya(aj . . . a„) 

 <?«! . . . dOn be the chance that they shall simultaneously lie between the limits 

 W] . . . «i + da^, &c. Let /i,(6, . . . b„) have the corresponding meaning for another 

 set of correlated variables independent of the a's, and so on toxfc, &c. 



Then, if we form the linear functions 



Xiffj + Xoi, +&c. 



the chance that they shall lie between the limits 7\ and 7\ + dr^ &c. is, by 

 Proposition I., 



Tco Tco 



\de, . . . \d6n^a1 

 J-<^ J -co 



Xge-"-'«'+ ••• ^'n'nW-i 



multiplied by a constant independent of r, . . . r„. 



8. In this expansion substitute for X^ &c. their values, and expand the 

 exponential factor. 



If any 6 differs from zero, eacb of the factors X is, by Prop. II., less than unity, 

 and therefore (since we are dealing not with the product of integrals, but with 

 the integral of the product X^ . . . Xj) we may neglect in that expansion all powers 

 of 6 above the second. The expansion is then reduced to the form 



II 



. . . €-'^d0j^ . . . d6„ 



