260 Mr. W. H. L. Russell on [Jan. 19, 



series A , A + A 1 , A -f-A 1 + A 2 + . . . Then by a second addition we 

 have A , A 1 +2A 03 A 2 + 2A 1 + 3A , or B , B ls B 2 , . . . 



Moreover, if the series A , A 1? A 2 , &c, is convergent, the series B,. 

 B l5 B 2 , &c, is also convergent, because 



B il+l _ A !l+1 + 2A B + 3A a . 1 +4A w . 3 + . . . 

 B ; , A„+2A w _ 1 +3A m _ 2 +4A^ 1 + . . * 



and if A ii + 1 is in the limit less than A M , A n less than A^ . . . B^ 

 mnst in the limit be less than B w . 



Hence if the limits are greater than a, and the difference between 

 either of them and a small when compared with (a), we shall have in 

 many cases excellent convergence. 



This method of converging to the valne of definite integrals is useful 

 in dynamical problems. Another is as follows. 



Let (p(x) = a + bx + cx 2 + ex* + . . . rx n =z%, 



then 0'(a)— =2z, 



dz 



dz 2 dz* 



dz 6 dz dz* dz 6 



When z=0, x is one of the roots of a + bx + cx--^ . . ,+rx n =0 ; call- 

 it O). Then 



dx =0 d?x 2 



dz ' dz* 0'(*)' 



fe_ Q fe__ 120"(*) 



Hence J——^tjL. . i_4- . . . , 



a (0 a)° 2 



jl+ ... 



J 30 a (0 a) b 2 . 5 



Bnt f xdz=xz—f zdx. 



Hence f dx s/ ' a + bx + cx 2 + e& 3 + . . . + ex n 



] • (227). 

 30'* (0*) 3 2.5 J V ' 



The limits are arbitrary ; bnt the series evidently requires that 0'(*) 

 should be considerable. 



Since 



^dxx^=\ 

 Jo n 



