Dr. J. H. Jellett on Laplace's Equation. 403 



If this expression be restricted to the first power of ecu, the 

 expression thus limited, namely 



r 



*uf(b) sin 6 d0d<j>, (6) 



JO 



represents the potential of a superficial distribution whose 

 density is ait. This is the form adopted by Laplace. The error 

 in his result is therefore of an order not lower than a 2 , provided 

 that no one of the terms u 2 f r (b) sin 6, u z f fr (b) sin 0, &c. become 

 infinite within the limits of integration. 

 In the present case 



/( r ) = r\r 2 + b 2 - 2br cos flfr = r 2 {(r-b cos Of + b 2 sm 2 0\ t\ 



Assume x = i — b cos 0, y — b siu 6, z = x* + y 2 . Then 



f(<r) = r 2 z m , putting m= — ~ — 

 We have then to inquire whether any term such as 



u p+i — sm (j 



drP 



becomes infinite when r — b. But as neither r*or its differen- 

 tial coefficients can become infinite, it will be sufficient to 

 determine the conditions necessary in order that 



nP +l - — sm0 

 dr p 



may remain finite when r = b, and therefore = 0. Now, since 



we have 



dP, 



elz _ 9 dx _ 

 dr ^ dr 



z=Az m -9 + Bz m - < i- 1 x2 + Cz m - ( i- 2 x i + &c, 



dr 

 or 



= AV""?-' x + Wz m ~«- 2 a? + &c, 



according as p = 2q or = 2q + ]. In both these series, since z 

 and x are both of the order 6 2 , when 6 is small, it is easily 

 seen that the first term is the term of the lowest order in 6. 



Moreover, as u = and -=^ =0 when 6 = 0, it is plain that u is 



also of the order 6 2 when 6 is small. Hence when jp = 2g, the 



order of the term u p '^ — = — sin is 

 ar p 



4q + 2 + 2m—2q + l = 2q + 2m + 3 = 2q + n + 4:; 



