100 Prof. C. Niven on the [May 29 r 



Class II. Any one of the forms 



(& s(Q+2(-i)^ . (2m + 25 ^_ 3 * S (2 ^ + 2 ; ^ 2 5 5 } + 1} (27 ^ + 4s + 1} - s + (f)) 



&SM + 2& 2S ' ( m + 1 )( m + 2) . . . (m + 8) „ n \ 



o Wf.. * -( 2m +2s+3)(2m+25H-5) . . (2m + 4s + l)" 



JH+l / 

 +1 



6 P m (f)+S(-l)'6 s P,(r). 



It is shown that the roots of ci^=0 are all real and definite in posi- 

 tion, and that in the neighbourhood of these roots the series a r , 

 . . . rapidly converge ; similar results hold for the roots of b ^ = 0. 



The general form of the ratios a r : a is ascertained, and the equation 

 <~'- cc = () written in the form 



l-e^ + ^Sg- ... =0. 

 where 2 r denotes the sum of the products of every r of the series 



^° , ... of which no two in the same product are adjacent. 



0o0i 010-2 



We then approximate to the values of z by series ascending by 

 powers of e, and the expression for the ?- + lth root is, up to e 5 , given 



by 



V0 r-1 r+l/ V0 r-20 \-l "r+10 r+ 2 / 



in which 0'^ represents the result of putting in <fip a first approxima- 

 tion to z, up to e given by (fir—0, and 0/' the resulting of substituting 



a second approximation up to e 3 given by r =e 2 ( ). 



These expressions are then used to expand z in powers of e ; and, 

 more especially, the first roots of each equation for any given value of 

 ru, are found. For the r + 1th root the term a r is the leading term of the 

 series, and expansions are found for a r+1 : a r , a r _ 1 : a r , a r+2 : a r . . . . 

 as far as e 3 . 



In the case of Class I a few of the lower roots are found for the 

 smaller numbers, m=0, 1, 2, and the numerical expansions given for 

 one or two of the coefficients adjacent to the leading one to (in general) 

 two terms. In Class II the numerical calculations have been performed 

 for a few of the roots only. These expansions represent the solution 

 found by M. Mathieu. 



The values of A, must be determined from the condition satisfied at 

 the surface of the solid. If the surface be maintained at constant 



