352 
MR. WARREN ON GEOMETRICAL REPRESENTATION 
For this may be proved nearly in the same manner as the preceding article. 
40. Let u — where x and m are any quantities whatever, and let in 
remain constant whilst x and u vary, 
The,i '^- = ” i O • 
For, since u = 
u' = in x' 
... . l d u l 
••• (by Art. 4.) - . Tx = m . 
du u 
dx x 
/ \ t 
= rn 
in 
(?) 
m — 1 
4 1 . Let z = 1 + x, and let z be inclined to unity at an angle = 6, 0 being 
positive and less than c, and let x be in length less than unity, and let m be 
any quantity whatever ; 
1 + in x + m ' 0 -- x + &c. | , if 6 be less than ”, 
tl,en C)“ = (!.)” ' 1 
m , 
. m — 
1 
1 .2 
m 
. m — 
1 
For, first, let 0 be less than j 
and let 
ey 
= A + Bi + Ci + &c., 
then differentiating 
“ (?) 
in .in — 
5 
m — 1 
■(;) 
&c. 
now let x = 0, 
= B + 2 C x + &c., 
= 2 C + &c., 
= &c. ; 
