186 



PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



whence we get for the formulae answering to those of Art. 22, 



M=l + 



16 



99 

 512 



x~ 



3 75 3 5715 



tan y = x ! + x~ 3 — . 



r 8 512 16384 



v 3 v _, s v - 3 "79 „ . 



x = X X *+ X 3 + JT -5 , 



8 128 5120 



^T being in this case equal to (i + 1) ir. 



Reducing to decimals as before, we get for the calculation of v for a given value of x, 



M = 1 + .1875ar 2 + .193359ar 4 , 

 tan \|/ = - .375j? _1 + ,146484a;- 3 - .348817a? _s , 

 (2x 



and for calculating the roots of the equation v = 0, 



x . .151982 .015399 .245835 

 — £ j. g« _ i. 



TT 4i + l (4i + l) 3 (4i + l) 5 



2x\i 3ir 



M cos (a? • - \|/) ; . . 



(63) 

 (64) 



(65) 



(66) 



25. The following table contains the first 12 roots of each of the equations u = 0, and 

 x~"v = 0. The first root of the former was got by interpolation from Mr. Airy's table, the 

 others were calculated from the series (56). The roots of the latter equation were all cal- 

 culated from the series (66), which is convergent enough even in the case of the first root. 

 The columns which contain the roots are followed by columns which contain the differences 

 between consecutive roots, which are added for the purpose of shewing how nearly equal these 

 differences are to 1, which is what they ultimately become when the order of the root is 

 indefinitely increased. 



