178 PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



12. The expression (28) will be rendered more easy of numerical calculation by assuming 

 R = M cos \|/, S = M sin \|/, and expanding M and tan -fy in series to a few terms. These 

 series will evidently proceed, the first according to even, and the second according to odd 

 inverse powers of (p. Putting the several terms, taken positively, under the form 1, (Mb -1 , 

 ab<p~ 2 , abcd>~ 3 , abcdfp' 4 &c, and proceeding to three terms in each series, we get 



M=l-a(k--)(p- 2 +albc(d-a) + -(b--]\<p- i , . . (31) 



tan >|/ = a<f>~ 1 - ab (c - a)<p~ 3 + ab\cd (e - a) — ab (c - o)|0 -5 . . . (32) 



The roots of the equation W = are required for the physical problem to which the 

 integral W relates. Now equations (28), (29) shew that when m is at all large the roots of 

 this equation are given very nearly by the formula <p — (i — £) ir, where i is an integer. 

 From the definition of \j/ it follows that the root satisfies exactly the equation 



<t> = (» - i) * + +• • • • - • • • • (33) 



By means of this equation we may expand <p in a series according to descending powers of 

 $>, where <£ = (i — ^) ir. For this purpose it will be convenient first to expand \j> in a series 

 according to descending powers of <p, by means of the expansion of tan -1 a? and the equation 

 (32), and having substituted the result in (33) to expand by Lagrange's theorem. The result 

 of the expansion carried as far as to <$>~ s is 



<p = $> + a<t> -1 - [ab(c - a) +^a 3 + a 2 } <I> -3 



+ {ab[cd(e-a) - ab (c - a)] + a 3 b{c-a) +^a 5 + 4a[a& (c-a) + ^a 3 ] + 2 a 3 } $~ 5 . ... (34) 



13. To facilitate the numerical calculation of the coefficients let 



a b c 



a = T7# ; 6 = 27z> ; c = sTd ; &c -' 



and let the coefficients of <A~ 8 , d>~* in (31) be put under the forms , - 



r r v ' ' 1.2Z) 2 1.2. 3. 4,D*> 



and similarly with respect to (32), (34). Then to calculate W for a given value of m, we have 



W = 2l(3m)-l M cos (<p --->//) , (35) 



where 



-W-l- -^ , ™0" , + Ai ^4 0-S (86) 



l.ZD 2Y 1.2.3.4Z) 4r V ' 



tan^-A-0- 1 ^__0-3 + 9* <t>-\ . . ( 37 ) 



r l.D Y l.Z.SD 3 ^ 1.2. 3. 4. 527^' V ' 



and for calculating the roots of the equation W = 0, we have 



d,= $ + -^L$-' ^t_^*-» + -> _$-» . . . (38) 



r l.D 1.2.3 2> 3 1.2.3.4.5D 6 V/ 



