570 LAPLACE. 



This will be found to agree with Laplace's page 311. 



For an example suppose that the function of the facility of 

 error is a constant, say C; then since 



[/(z) dz = 1, 



•^ 



we have aC =1. 



2 _2 



Thus ^' = 9 » ^' = -o ' ^*^ ~" ^"^ = 19 • 



Therefore the probability that the sum of the errors will lie 



T sa ^ sa . 



between -^ — v and -^ + t; is 



1006. Laplace next investigates the probability that the sum 

 of the squares of the errors will lie between assigned limits, sup- 

 posing the function of the facility of error to be the same at 

 every observation, and positive and negative errors equally likely. 

 In order to give the result we must first generalise Poisson's 

 problem. 



Let 0i {z) denote any function of z : required the probability 

 that 



will lie between the limits c — rj and c + 77. The investigation 

 will differ very slightly from that in Art. 1002. In that Article 

 we have 



J b 



g7,a;W-i^^ 



in the present case the exponent of e instead of being y^xzj—l, 



will be x(pi {z) J— 1. The required probability will be found 

 to be 



- — -r- e 4"' dv; 



