J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 333 



body may also be expressed by a simple integral, if we imagine 

 it decomposed into a series of infinitely thin cylindrical shells, 

 all perpendicular to the plane of the x, y. If we make 



r2 = 07- + y'^, 

 the solid contents of every such cylindrical shell of infinitely 

 small thickness will be found 



= 2r zv. dr, 

 consequently, the volume of the body (for which, in relation to r, 

 we must take the limits to co ) ; or 



t/ 



Qrire dr, 



for which the integral is immediately found, 



■n I d ( — < 

 *J 



V = 7r / d{-e 



or for the given limits, 



V = 7r. 



Hence, by virtue of the above, 



L = Vt; 



and consequently, by substituting this value in (5.), 



X 



-^ V'^r = 1, 

 or 



- J_ 



The complete expression for 4) A will be accordingly 



:? A = -^e-'''^% (6.) 



which not only contains in itself the principle of the arithme- 

 tical mean, but depends so immediately upon it, that for all those 

 magnitudes for which the arithmetical mean holds good in the 

 simple cases in which it is principally applied, no other law of 

 probabihty can be assumed than that which is expressed by this 

 formula. It is therefore not limited to any special kind of ob- 

 servation, but is altogether general. Equally general is the re- 

 sult in regard to fl, which follows immediately from this form : 

 namely that,ybr any arbitrary nwnber of magnitudes to be deter- 

 mined, the most probable values are those which make the sum 

 A^ A* + A'- a" + A"' A"'^ a minimum. 



