

I 



ON GAUSS S THEOREM FOR QUADRATURE, ETC. 389 



and therefore 



^{"Ax)dx=lJ5fic,') + 8fic,) + 5/ic,)], 



where 



c^' ^^{p + q)-h{p-g) {r}'=Up + q)--38Tdip-q), 



c, =HP + q) + h{p-q) i^y = hip + q) ^-3873 ip-q). 

 As an example, let it be required to find the value of 



2 



approximately. We have 



1 2 



e"-^ dx 



c,' = -1127 , c',2=-0127, e-^' =-9874, 



Co =-5 , Co2=-25 , e-'^" =-7788, 



_c 



•2 



then 



Ci =-8873 , c,- = -7873 , e-'^' =-4551 ; 



rV(5e-^'' +8e-^" +56"'''') = -7468. 



2 

 Multiplying by—, we find as the approximate value -84268. 



The value of 



2 f _a;' 



e dx as given in the tables is -84270 ; the ap- 

 proximation in this example is closer than in other examples and must 

 be affected by the special form of the subject of integration. 



P II. Let 7Z=5, so that p=2. The quantities a,- and a^- are the 

 roots of 



and we take 

 Further, 



63ii2_70/; + 15 = 0, 

 63a, 2 = 35 + (280)*, 63a/--=35 - (280)4 



Ao= J-^{i-i(a,2 + a,2)+a.wl 



^128 



225' 



"I (ai — a: ) 



lol yi __ .OQcq 



~45d 45(280)* 



, 161, 91 .r,n.. 



1919. • G G 



