LAPLACE. 545 



a single observation to be known : Laplace's formula when applied 

 by him to a special case coincides with that which we have given 

 in Art. 567 from Lagrange. 



989. An example is given by Laplace, on his page 271, which 

 we may conveniently treat independently of his general investi- 

 gation, with which he himself connects it. Let there be a number 

 n of points ranged in a straight line, and let ordinates be drawn 

 at these points ; the sum of these ordinates is to be equal to s : 

 moreover the first ordinate is not to be greater than the second, 

 the second not greater than the third, and so on. Required the 

 mean value of the r*^ ordinate. 



Let z^ denote the first ordinate, let z^ + z,^ denote the second, 

 ^1 + ^2+ ^3 ^^^ third, and so on : thus z^, z^, z^, ... z^ are all posi- 

 tive variables, and since the sum of the ordinates is s we have 



nz^-^{n-l)z^-\- (n-2) z^+ ... +z,, = s (1). 



The mean value of the r*^ ordinate will be 



{z^-{-z^+ ... +Zr) dz^dz^,.. dz^ 



ctz^ (^z~ . . . CIZ^^ 



where the integrations are to be extended over all positive values 

 of the variables consistent with the limitation (1). 



Put nz^ = x^, (ii — l)z^ = x^, and so on. Then our expression 

 becomes 



... p + — ^H--^ + ... -\ ^^^--^]dx^dx.-^...dx,, 



JJJ \n 71 — 1 71 — z 7i — r-\-l/ 



> 



... CvX^ (JuJUn ... Ct/ll/j^ 



with the limitation 



x^ + x^+ ...+x„ = s (2). 



The result then follows by the aid of the theorem of Lejeune 

 Dirichlet : we shall shew that this result is 



s (1 1 1 ,11 



n \n 71 — 1 71- z 7i — 7'-\-l) 



35 



