194 Mr. A. Cayley on an Analytical Theorem connected with 



then 



«/>(/i, ar) = AoPo + A,Pia? + AgPs^^ + &c., 



where Pq, Pi, Pa, &c, are Legendre's functions, viz. the functions 

 of /J, which are the coefficients of the successive powers of x in 

 the development of (1— 2/Xcr + .r^)"* in ascending powers of x. 

 And the electrical thickness y at anjf point of the surface of the 

 sphere radius 1 is given by the formula 



where, after the diflFerentiation, « = !. 



The problem conseqviently depends on the determination of 

 the potential /a: for a point on the axis; and this is determined 

 by the functional equation 



. b / l + b-x \ _ bh 



^^ l + 26-(l + %''\l + 26-(l + %y 1 + 6-a; 



(Plana's equation (G), in which I have written for /8, 7, H their 

 values, and substituted also for g its value =A). The solution 

 of this equation is (equation (H), writing therein g-=h) 

 P 00 1 



^"^ = T^:^ + ^^^"0 6 + n(l + A)-M(l + % 



^« 1 



~^^""o(« + l)(l + 6)-(l + n(l+A)y 



where P is an arbitrary constant quoad the functional equation, 

 viz. any function whatever which has the property of remaining 



\ _i_5 X 



unaltered when x is changed into , . „, — ,-,,,, . Poissonjand 

 ° l+2o— (l + oja; 



Plana after him, arrive at the conclusion that in the physical 



problem P = 0. It appears to me that there is ground for hold- 



P 



ing that this is only true sub modo, and that . .^ for a?=l 



(which, if P were retained, would be a term occurring in the ex- 

 pression for the thickness at the point of contact) is not of neces- 

 sity zero. But the term, if it exists, can be replaced at the con- 

 clusion ; and I write therefore 



fx = bh2n ;. .„na.M_«n^M^ - *^'^» 



ob + n{l + b)-n{l + b)x ^\(ji + i)(i + b)-(l+n{l +b)')x'' 



According to the process by which the solution of the func- 

 tional equation was obtained, this is the true form of the solution ; 

 for although the series are non-con^'ergent, and the two sums 

 are in fact each of them infinite, there is nothing to show a rela- 

 tion between the number of terms which must be taken in each 



