180 PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



dW 



16. To determine when IF is a maximum or minimum we must put = 0. We might 



r dm 8 



dW 



get by direct differentiation, but the law of the series will be more easily obtained from the 



dm 



3 TT 



differential equation. Resuming equation (ll), and putting V for — - , we get by dividing by 



n and then differentiating 



d?V \dV n„ 



+-F=0. 



dn 2 n dn 3 



This equation may be integrated by descending series just as before, and the arbitrary 

 constants will be determined at once by comparing the result with the derivative of the second 

 member of (15), in which A, B are given by (24). As the process cannot fail to be understood 

 from what precedes, it will be sufficient to give the result, which is 



where 



V=3-i7rinUR'cos (<p + -) + 6" sin (cf> + -jl, .... (43) 



, -1.7.5.13 -1.7.5.13.11.19.17.25 



~~ 1.2(720) 2 1.2.3.4(720)* 



, - 1-7 - 1.7.5.13.11.19 

 " 1.720 1.2.3(720) 3 



(44) 



17. The expression within brackets in (43) may be reduced to the form M cos (0h \j/) 



just as before, and the formulas of Art. 13 will apply to this case if we put 

 o'- -1,7: 6'= 5.13; c'=ll.l9; &c, D = 72. 



dW 

 The roots of the equation - — = are evidently the same as those of V = 0. They are given 



approximately by the formula = (i - £ ) tr, and satisfy exactly the equation = (i-fyir + yj/. 

 The root corresponding to any integer i may be expanded in a series according to the inverse 

 odd powers of 4i-3 by the formulae of Art. 13. Putting (t- A) 7r for 4>, and taking the 

 series to three terms only, we get 



Ei m - 7 ; E t m - 84168 ; 

 whence 



. 7 H69 . , 



T 72 31104 



or, reducing as before, 



^..^-f^+^^L (45) 



TT 4* - 3 (4i - 3) 3 



This series will give only a rough approximation to the first root, but will answer very well 

 for the others. 



For i = 1 the series gives 7r _1 ■= .25 - .039 + 025, which becomes on taking half the 

 second term and a quarter of its first difference .25 - .019 - .004 = .227, whence m = 1.12. 



