344 
MR. WARREN ON GEOMETRICAL REPRESENTATION 
then V = m a ! ; 
X p 
In like manner let V = m a ' ; 
^ 7 
then V — b' = m a' — m a', 
X y p q 
(by Art. 7.) x — y . c J — 1 = m.p — q.c*J — \, 
.'. x — y — m . p — q, 
where m is either irrational or impossible ; therefore, since x, y, p, q are either 
= 0 or are whole numbers either positive or negative, the conditions of the 
equation cannot be satisfied unless p = q, 
p = q. 
21. Let (a)- = b, and let (b\ n — f, where a, m, n, are any quantities 
C) 
whatever ; 
then, if V be that value of b’, which is equal to m a', (a\ mn — f- 
? p \pJ 
For, since f = ^b^ n 
f — nV 
= m n a! 
p 
m n 
•*•/= ( a 
(\ 7 
D 
22 
• (?) 
m n — 1 
py where m and n are any possible quanti- 
ties whatever, and c is the circumference of the circle, and E the base of the 
o 
hyperbolic logarithms. 
For let ^E\" + " v ^ i = g, 
then one of the values of g' is m + n ,J — 1, 
let g' be that value of g', 
‘ i 
then g' = m + n ^ — 1 ; 
7 
let r be a positive quantity, in length = g, and let g be inclined to unity at an 
angle = 0, 0 being positive and less than c the circumference of the circle, 
then g' = r' -\- 0 + q c . *y — 1; 
