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



The equation (2.) 



/ 



+ 00 



<fi A c? A = 1 



may then serve for the further determination of a constant. If 

 we make h A = t, this integral becomes 



•+CD 



. ... .r 



(5). . . . I / e-''dt = h 



where the limits remain the same as before. 



In order to obtain the value of this definite integral, let us 

 examine the double integral* 



+ 00 



^'fj 



'dxdy, 



where x and y signify two variable magnitudes independent of 

 each other, and the limits — co to + oo refer to the integration 

 according to x as well as to that according to y. If we integrate 

 first according to y, considering x as constant, and make the 

 value 



>+oo 



? ~ ^ dy = L, 



/: 



then 



"+« 



='/- 



e dx 



consequently, if we now integrate according to x, 

 V = U. 

 But we may also compare the expression for V with the ge- 

 neral formula for the oubature of a solid. If we consider x, y, z 

 as three rectangular co-ordinates, and imagine the surface the 

 equation of which 



. ^ g - (^^ + y^) 



V will express the volume of the body bounded by this infinitely 

 extended surface. But this surface would obviously have arisen 

 by rotation round the axis of the z, because z comes alike to all 

 points of the plane of the x, y, which are equally distant from 

 the point of beginning. On this account the volume of the 



• According to the verbal communication to which I am indebted for this 

 short and elegant mode of finding the value of the definite integral, M. Cauchy 

 has given it thus in his lectures. 



