20 GENERAL PRELIMINARY RESULTS. 



LAPLACE has shown, in his Me*c. Celeste, that the function F 

 has the property of satisfying the equation 



~ dx* dtf dz z ' 



and as this equation will be incessantly recurring in what 

 follows, we shall write it in the abridged form = S F; the 

 symbol S being used in no other sense throughout the whole of 

 this Essay. 



In order to prove that = 8 F, we have only to remark, that 



by differentiation we immediately obtain = S , and conse- 



quently each element of F substituted for F in the above equa- 

 tion satisfies it ; hence the whole integral (being considered as 

 the sum of all these elements) will also satisfy it. This reason- 

 ing ceases to hold good when the point p is within the body, 

 for then, the coefficients of some of the elements which enter 

 into F becoming infinite, it does not therefore necessarily follow 

 that F satisfies the equation 



although each of its elements, considered separately, may do so. 

 In order to determine what 8 F becomes for any point within 

 the body, conceive an exceedingly small sphere whose radius 

 is a inclosing the point p at the distance b from its centre, a and 

 b being exceedingly small quantities. Then, the value of F 

 may be considered as composed of two parts, one due to the 

 sphere itself, the other due to the whole mass exterior to it : but 

 the last part evidently becomes equal to zero when substituted 

 for F in 8 F, we have therefore only to determine the value of 

 8 F for the small sphere itself, which value is known to be 



p being equal to the density within the sphere and consequently 

 to the value of p at p. If now x t , y t , z,, be the co-ordinates of 

 the centre of the sphere, we have 



