168 MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
equations in finite terms would be practicable in very few cases. The following 
method of determining a well-known definite integral is here added, to show the con- 
nexion between previous investigations relative to definite integrals, and those given 
in the present memoir. 
We know that 
or 
.(2r)M (2rr .1.3_ (2rf 1-3.5,, ^ 
1221.234 2^ 123456 2^ * 
Hence remembering that \ dz 
we find 
j: 
j: 
g cos 2 r; 2 =-^g 
I shall now enter on some investigations connected with Lagrange’s theorem. 
Let 1 —«/-!-«?/’■ =0 be an algebraical equation. Then Lagrange’s theorem gives us 
the following series : — 
m(m+ 2r’ — 1) 
m[m + nr—\) (m + nr— 2) ....(m + n(r— 1) -t- 1) 
— ^a^-f&c.-l i:07. 7? r a”-f &c. 
If we apply the usual test of convergency to this series, we find that (r— l)a must 
be less than unity. 
Then we see that 
» -7= 1 +(m+2r- 1)«+^ «’+&c. 
, {m + nr—\) {m-\-nr—2)...{m-{-n[r—\)-\-\) ^ 
+ 1.2.3...(«-1) “ 
V{m + nr) 
Now (m-f/ir— 1) (m-l-wr-2)...(m-f w(r— = ; 
wherefore, since ^ cos“’*'*“^^ 
we have 
— \ ) — 1) + 1) 
J_JL 
Cos»»+«’->^ g(m+«(r-2) + l)i0^^ 
00 ^ 
^m+nr— 1 
Hence we have l-|-(m-|-2r— l)a-l- 
rtW»+r— 
{m + ^r — 1) (m-1-3?’ — 2) 
1.2 
a^-)-&c. 
t+r-l n-J 
^ Jo 
dMz COS”+’-‘^ 5 * ecos(r- 2 )e-. 
cos(2’'az cos’'^sin(r— 2)^-}-(m-l-» — 1)^) ; 
