:U2 
MR. WARREN ON GEOMETRICAL REPRESENTATION 
For, let v — general logarithm of § in a system whose base is E, 
0 
then (by Art. 1 1 .) v = j?, = y — §'• 
O 
14. Cor. 3.) Hence the quantity E in the preceding article is the base of 
0 
the system of general hyperbolic logarithms. 
15. Def.) Let a and m be any quantities whatever, and let £ be a quantity 
such that one of its general logarithms in a system, whose base is a, is m ; 
then f is called the m th general power of a, and expressed thus cT ; and the m th 
general power of a is expressed thus / a\ m . 
p \p/ 
16. Cor.) Hence = m a!. 
17. Let a be any quantity whatever, and let b be a positive quantity in 
length equal to a, and let a be inclined to unity at an angle = a, where a is 
positive and less than c the circumference of the circle, and let £> = (a^j n ; 
then, if m be a possible quantity, § will be in length = and will be in- 
clined to unity at an angle = m . ct p c . J — 1. 
For, since § = ( a\ w , one of the values of ^ is m a 1 , 
\p / p 
Let be that value of £>', 
then (} = m a! 
1 p 
= mb' + ?n . a + p c . — 1 ; 
O 
Let r be a positive quantity, in length = £, and let p be inclined to unity at 
an angle = D, 0 being positive and less than c, 
then §’ = r' + 0 -f q c . — 1, 
q 0 
.-. r' + 0 + q c . — 1 = m V -f m . a + p c . — 1, 
0 0 
r ' = m b' , and 0 + qc = m.a-\-pc 
0 0 
f is in length = (b\ ; 
