724 Mk. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



y' 



that it will be sufficient to suppose a? = .5. The value of z, T, or — for sc = .5 may be readily 



S 



calculated by the method explained in Art. 21. I have also obtained the following expression for 



this particular value ; 



= ^^ - ^'^ir^ - iTT^ -^ 3-iT^ - 4 



.(42) 



When j8 is small, or only moderately large, the series (42) appears more convenient for 

 numerical calculation, at least with the assistance of a table of reciprocals, than the series (39), but 

 when /3 is very large the latter is more convenient than the former. In using the series (42), it will 

 be best to sum the series within brackets directly to a few terms, and then find the remainder from 

 the formula 



W* - Mx+l + W*+^ - ••• = i"r - i Am, + 1 A'^M, - ... 



The formula (42) was obtained from equation (20) by a transformation of the definite integral. 

 In the transformation of Art. 8, the limits of s will be 1 and 03 , and the definite integral on which 

 the result depends will be 



X 



1 +■ 



ds. 



The formula (42) may be obtained by expanding the denominator, integrating, and expressing 

 m in terms of jS. 



In practice the values of /3 are very large, and it will be convenient to expand according to 

 inverse powers of/3. This may be easily effected by successive substitutions. Putting for shortness 

 jc — x"- = X, equation (4) becomes by a slight transformation 



aw 

 and we have for a first approximation y = X', for a second y = X'- - /3~'X^ ^ , and so on. The 



0/31 



result of the successive substitutions may be expressed as follows : 



y = X^-l^-^X^f^X^^^-^X^±^X^§-^X^-Uc., (43) 



where each term, taken positively, is derived from the preceding by differentiating twice, and then 

 multiplying by j3"'^". 



For such large values of (i, we need attend to nothing but the value of ar for x = ^, and this 

 maybe obtained from (43) by putting a' = ^, after differentiation, and multiplying by l6. It will 



however be more convenient to replace x by i(l + w), which gives — — = 4-—-; X^ = ^yv, 



where ir = (l - w'f. We thus get from (43) 



a = Ff - {i^Y'Wp, W + (i^y^W^^ W-p, W- ... , 

 aw dw dw 



where we must put w = after differentiation, if we wish to get the value of sr for a? = ^. This 

 equation gives, on performing the differentiations and multiplications, and then putting w = 0, 



« = 1 + /3-' + I/3-' + 13/3-'+ (44) 



In practical cases this series may be reduced to 1 + /3"'. The latter term is the same as would 

 be got by taking into account the centrifugal force, and substituting, in the small term involving 

 that force, the radius of curvature of the equilibrium trajectory for the radius of curvature of the 



