58 M. C. Cellerier on a Simplification of 



values of the excursion — that is to say, any given function/ 

 of the position of the point in space. For this we must have 



/=£a cos pp. 



This mode of representation of a function is Fourier's formula. 



The values of the unknowns thus found are sextuple inte- 

 grals, an exact solution of the problem, but which give no idea 

 of the general form of the motion. It is only after laborious 

 transformations, supposing the initial disturbance included 

 within a limited space, that we arrive at interpreting them so 

 as to make evident the limited wave-form. 



Beside this complication, Fourier's formula has another in- 

 convenience: the integration with respect to some of the vari- 

 ables has, if we commence with them, an indeterminate result; 

 and differentiation under the symbol of integration offers but 

 little guarantee of accuracy. 



Now these inconveniences may be avoided by substituting 

 for Fourier's formula another, likewise representing an arbi- 

 trary function f(x, y, z) of three indeterminates which may be 

 regarded as rectangular coordinates of a variable point : the 

 function is supposed =0 if the point is outside of a limited 

 space designated by V. For the enunciation of the formula, 

 we will denote by S a spherical surface having unity for radius, 

 and for its centre the origin ; it shall be divided into ele- 

 ments ft) ; designating by H the position of any one of them, 

 «, /3, 7 will be the cosines of the angles made by the straight 

 line OH with the axes ; P will denote a plane perpendicular 

 to OH, cutting this straight line at a distance p from the 

 origin, which distance will be taken as negative on the side 

 opposite to H ; lastly, we will put 



F(p)=$jlx',y',z')<o', 



the summation extending to all the elements ft/ of the plane P, 

 and x r , y', z' being the coordinates of each. 



Thus F(p) will be a function solely of p and the cosines 

 a, /3, y. Let <j>(p) be its second derivate with respect to p, in 

 taking which a, fi f y are regarded as constants. The following 

 is the formula sought : — 



A a > V**) = - g^2 2(«0 + Py + V*)<o, 



in which the symbol of summation extends to all the elements 

 ft) of the sphere S ; &, /3, y correspond to each ; x, y, z are the 

 coordinates of any point in space. 

 The preceding expression comprises in the main four inte- 



