CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 179 



The coefficients in these formulae are given by the equations 



At = a (b'-a) ; A, = a \b'c' (d'- 4a') + 3a' 8 (2b' - a')\;) 

 &*«'; C 3 = a'b' (»' -3a); C 5 = a'b' \c'd'(e'- 5a) - 10C 3 } ; 



£,-«'; 2J 3 = C 3 + 2a' 2 (3JD + a'); 

 £ s = C 4 + 20a' (42? + a') C 3 + 24a' 5 + 80a' 3 D (3D + 2 a'). 



(39) 



14. Putting in these formulae 



a'=1.5; fe' = 7.H; c'= 13,17; a"=19.23; e' = 25.29; 2) = 72; 



we get 



4, = 5.72; A t = 3. 5. 72 s . 457; d = 5 ; C 3 = 2 . 5 . 7 . 11 . 103 ; 

 C 5 = 4 2 .5 3 .7 2 . 11.23861; ^ = 5; 2? 3 = 72 . 1255 ; E b = 4 . 5 3 . 72 2 . 10883 ; 

 whence we obtain, on substituting in (36), (37), (38), 



5 _ 2 2285 



ilf=l- a>~ 2 + d>-\ 



144 r 41472 r 



5 39655 321526975 6 



tanvi/- = — l 3 + , 



r 72 r 1119744 r 2902376448 r 



(h = $ + — (J) -1 $ 3 H 4>~\ 



r 72 31104 2239488 



Reducing to decimals, having previously divided the last equation by v, and put for 4> its 

 value (i - ^) ir, we get 



M= 1 - .034722 <p~* + .055097 0~ 4 , (40) 



tan \J/ = .069444 (p- 1 - .035414 -3 + .1 10781 <p~ b , . . (41) 



(h .028145 .026510 .129402 



i - i - AS + — — .. + — ... (42) 



7T 4t - 1 (4» - l) 3 (4l - l) 3 v ' 



15. Supposing i = 1 in (42), we get 



2- = .75 + .0094 - .0010 + 0005 = .7589 ; 



7T 



whence m = 3 I — ) = 2.496. The descending series obtained in this paper fail for small values 



of m ; but it appears from Mr. Airy's table that for such values the function W is positive, 

 the first change of sign occurring between m = 2.4 and m = 2.6. Hence the integer i in (42) 

 is that which marks the order of the root. A more exact value of the first root, obtained 

 by interpolation from Mr. Airy's table, is 2.4955. For i = 1 the series (42) is not convergent 

 enough to give the root to more than three places of decimals, but the succeeding roots are 

 given by this series with great accuracy. Thus, even in the case of the second root the value 

 of the last term in (42) is only .OOOOO7698. It appears then that this term might have been 

 left out altogether. 



23—2 



