14 Mr. S. H. Burbury on the Law of Probability 



Also i£ A denote the determinant 



A== 



/3 2 l «2 P23 



(20) 



or 



A = 2 + 



du t du x 

 di\ dr 2 



then by a known proposition A = ^ . And as 



so reciprocally 





An 

 A ' 



/3i2=-p-, &c., 



&12== ^2,&C. 



These results may be confirmed as follows : — 1£ we integrate 

 r^e"^ for all the variables r except r x between limits ±00,. 



we obtain r 1 2 =— ^. But r 1 2 = a 1 by definition. Therefore 



a x =~-^. Similarly Vy 2 = b 12 =^-~ , and so on. In the 

 way, expressing Q in terms of the u's, 

 — Q 2 7 



same 



Ji- 

 ff 



,-Q 



,7 D ^ 



9/ 2 



u i j 



. . . <? ^L^tlodUz . . . <^ = -pr- = /5i 2 = ?«i^ 2 &C. 



■Q fl 



D 



2r0V-l 



24. The integration of g"* *-■»■*-* for ^...ft, between 

 limits +co is given by Todhunter (Cambridge Phil. Trans, 

 vol. xi. 1871, p. 219) on the assumption thai the coefficients 

 in Q are such as to make Q e necessarily positive. The 

 necessary and sufficient condition for which is that D and 

 every coaxial minor of D is positive. 



Let $. . je 1 . . .de n e-^ . *- 2rt ^=u. 



Then replacing e~ lrd ^ _1 by cos%r0 — \/ — 1 sinSrtf, we have,, 

 since the imaginary term disappears in the integration 

 between -f- co and — go , 



U =jj. . .dOi. . .d0 n e- qe cos 2>0. 



