184 PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



For example, when a? = 10 we have, retaining 5 decimal places in the series, 

 R = 1 - .00070 + .00001 = .99931 ; S = .01250 - .00010 = .01240 



Angle w - — m 527°. 95780 = 3 k 180° - 12°2'32" ; whence u = - .24594, 



which agrees with the number (- .2460) obtained by Mr. Airy by a far more laborious process, 

 namely, by calculating from the original series. 



22. The second member of equation (52) may be reduced to the same form as that of (28), 

 and a series obtained for calculating the roots of the equation u = just as before. The for- 

 mulae of Art. 13 may be used for this purpose on putting 



a' = l 2 ; b' = 3 a ; c' = 5* ; &c. ; D = 8, 



and writing a?, X for (p, <I>, where X = (i — £) ir. We obtain 



A 2 = 8 ; A i = 3 . 8 2 . 53 ; C, = 1 ; C 3 = 2 . 3 8 . 11 ; G 5 = 3 2 . 4 2 . 5 . 1139 ; 



JS", = 1 ; £" 3 = 8 . 31 ; E, = 4 4 . 3779 ; 

 whence we get for calculating u for a given value of ,v 



16 512 ' 



. 1 , 33 _ 3417 , 



tan\l/ = -# * cc 3 + w~ i , 



r 8 512 16384 



2 \* 



u 



-(-) if co. (.-J-*) (53) 



For calculating the roots of the equation u = we have 



8 384 15360 



Reducing to decimals as before, we get 



M m 1 - .0625a? -2 + . 103516a?- 4 , (54) 



tan \// = .125«t? -1 - .064453a?" 3 -t- ,208557a?" 5 , . . (55) 



a? . -050661 -053041 -262051 , : 



- = % - -25 + + . . (56) 



tt 4i-l (4i-l) 3 (4i-l) 5 V ' 



As before, the series (56) is not sufficiently convergent when i = 1 to give a very accurate 

 result. In this case we get 



TT-'a? = .75 + .017 - .002 + .001 = -766, 



whence a? = 2.41. Mr. Airy's table gives u = + -0025 for a? = 2*4, and u = — -0968 for x = 2 - 6, 

 whence the value of the root is 2.4050 nearly. 



The value of the last term in (56) is .0000156 for i = 2, and .00000163 for i = 3, so that all 

 the roots after the first may be calculated very accurately from this series. 



