SECT. 3.] ANY NUMBER OF OBSERVATIONS. 265 



III. By putting again, 



i^ = / = r + /V + &amp;lt;r*-fetc., and W ^= W&quot;, 

 we shall have 



/=*_!/_/ j-, 



also W&quot; independent of p, and q, as well as r. Finally, that the coefficient of y&quot; 

 must be positive is proved in the same manner as in II. In fact, it is readily per 

 ceived, that y&quot; is the sum of the coefficients of rr in vv, v v , v&quot;if , etc., after the 

 quantities p and q have been eliminated from v, v , v&quot;, etc., by means of the equa 

 tions /== 0, q = 0. 



IV. In the same way, by putting 



we shall have 



&amp;gt; n i i j 



iS n n r 



f^j p .1 i/ i/ i 



a P / 



W&quot; independent of p, q, r, s, and 8 &quot; a positive quantity. 



V. In this manner, if besides p, q, r, s, there are still other unknown quanti 

 ties, we can proceed further, so that at length we may have 







+ s s + etc + Constant, 



in which all the coefficients will be positive quantities. 



VI. Now the probability of any system of determinate values for the quan 

 tities p, q, r, s, etc. is proportional to the function e~ hhw ; wherefore, the value of 

 the quantity p remaining indeterminate, the probability of a system of determi 

 nate values for the rest, will be proportional to the integral 



fe~ hhW Ap 

 extended from jt&amp;gt; oo to p=-^-ao , which, by the theorem of LAPLACE, becomes 



therefore, this probability will be proportional to the function e~ hhw . In the 

 same manner, if, in addition, q is treated as indeterminate, the probability of a 



34 



