THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMTC PROBLEM. 237 



__ -_, v + 1+JL+ N B v /(fr - 3fc + 1)1 



2&"(6^+V,+ l)- 



M 5 - -- & 85 +13 68 + 176 209 + 43 + 20 + 305 359 235 + 298+413 



308 437+191+386 70 -260 3+130+21 44 11+9+26 1, 



N- = &-"'+! 14(5 + 84 13 23 100 + 215 + 59 !97 210 + 231 



+274 143 297 + 34+230+40 127 54+45 + 32 7 9?>3+0+l (13): 



S , _ (/>-!)<> {M, v /(/> 2 + & + IL+ N n v/(/r - 3& + 1)J 



ft 26 i!! (6 4 + 6 + l)s 



M fi = //'" + 15 93 + 299 494 + 298 + 70 + 387 I I 51 + 1 28 + I 2 1 (I 

 + 370 - 1762 - 619 + 1720 + 982 - 1:349 - 1086 + 77:! + 89 I 



288 540 + 36 + 232 + 25 65 13 + 11 + 'll \ . 



N fi = - & 2n +13 67 +165 166 10 56 + 513 285 588 + 26 + 1062 



+ 152 11(55 5(50 + 1 014 827 594 -815+1 69 + 562 + 64 



266 100 + 75 + 49-9- 11/r-f 0+1 (14). 



These calculations, as well as for ^ = 11 and 13, and their verification, were carried 

 out for me by Mr. J. W. HICKS, of Greenwich Observatory. 

 Putting 



M- " 2//' (//' + /> + I ) 

 then, since 



we find 



. 3 = /; u + 14 43 + 28 + I 9 + 22 54 - 1 2 + 30 + 22 17 8 + 5 + 21 (I 7). 



b 3 = (b 1) (h n - 1 2 + I !) + 1 1 - 9 1 9 + 5 + 1 5 1 -- 5 + + I ) (1 8). 



/ 1 \ 



The substitution (b, - ) changes v into 4w, 2r into 80, . . ., so that * and /> s are 

 obtained from a 2 and />., by writing the coefficients in reverse order. 

 tta - _ /,i i + 2 + 5 - 8 - 17 + 22 + 30 - I 2 - 54 + 22 + 19 + 28 - 43 + 1 4 - I (1 9), 

 6 S = _ (b - l) (6 n + - 5 - 1 + 15 + 5 - 19 -9 + 11 + 19-12 + 11 (20). 



Again, since 



12fW ,- v2 (T) /.7i\ 



, = (Jr 4K \^ l )> 



we find 



a 1= b u + 2 + 5-8-5-2 + 18 + 0-18-2 + 7 + 16-7 + 2-1 (22) 

 6, = - (b - 1) (b u + 0-5-1 + 3+5-7-9-1 + 7 + 0+1) (23), 



and in a~, b^ the coefficients run in reverse order. 



