402 Dr. J . H. Jellett on Laplace's Equation. 



sufficient that U remain finite for all values of and <£ included 

 within the limits of integration. Applying this principle to the 

 integral in (4) and (5), we see that the process of integration 

 will be legitimate if ep n+1 sin remain finite within the limits 

 of integration. Now it is evident that this expression can be- 

 come very great only when n + 1 is negative and p( = PQ) is 

 very small. But if p is small, it is plain that is small. It will be 

 sufficient, therefore, to consider the limiting value of ep n+] sin 

 for very small values of 0. Now e, the density of distribution, 

 is proportional to the thickness of the shell at each point. 

 Hence if a line be drawn from 0, the centre of the sphere, 

 cutting the sphere and the original surface in Q, Q' respec- 

 tively, the density of distribution, e, at the point Q, will be 

 proportional to QQ'. Now, observing that OP is a normal to 

 both surfaces, we see that, if POQ( = 0) be a small angle, 

 QQ' cannot be of an order lower than 2 . For both OQ and 

 OQ' differ from OP by a quantity of this order. Hence 

 the greatest value which e can have at the point Q is K<9 2 , 

 where K is finite. Again, p ■= 2bs'm^0 = b0 nearly, and 

 sin = 0, to the same order. Hence ep n+1 sin 0= K# ra+4 + 

 higher terms. It is evident, then, that ep n+1 sin will not 

 be infinite for (9 = unless w + 4<0. Laplace's equation is 

 therefore true for all positive values of n, and for all negative 

 values (except n= — 1) which are not numerically greater 

 than 4. The equation may be true for higher negative values 

 of n than —4, if the sphere have contact of an order higher 

 than the first with the original surface. 



It remains to determine the degree of approximation to 

 which the equation (2) is true. The order of the error in this 

 equation is the same as that of the assumption by which this 

 result was obtained, namely that a distribution on the surface 

 of the sphere may be substituted for the shell. 



Suppose that, in general, the potential of the shell is repre- 

 sented by the integral 



§§§f(r)dr sm Od0d<l>. 



Let 



Then, the limiting values of r being r=b, r = b + uu, the 

 potential will be 



fn fin 



v=\ \ \f l (b + cm)-f l {b)\sm0d0d<}> 



Jo J 



= y p" { uufb + i « V / 7 (b) + &c. } sin d0 dcf>. 



