342 MR. WARREN ON GEOMETRICAL REPRESENTATION ' * '^ 





For, let V = general logarithm of f in a system whose base is E, 

 then (by Art. ll.) v = ^, = \ = g'. ■ ' ' " 



14. Cor. 3.) Hence the quantity E in the preceding article is the base of 



the system of general hyperbolic logarithms. ^ ' 



15. Def.) Let a and m be any quantities whatever, and let f be a quantity 

 such that one of its general logarithms in a system, whose base is a, is m; 



then § is called the m"" general power of a, and expressed thus a"* ; and the »»*'' 



general power of a is expressed thus /a\ . < vi ■ 'i^ 



, 16. Cor.) Hence g' — m a'. 

 17. Let a be any quantity whatever, and let 6 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\ ; 



then, if m be a possible quantity, g will be in length = f *'\™ ^^^ ^i^^ ^^ ^^~ 



clined to unity at an angle = /«.a+pc.*y— 1. 

 For, since § = /^«\™, one of the values of §' is m a', 



• ^ **■• Let f' be that value of f', , '•'*^\ 



■* * ? - , •%' . " ,,.»'•, 



then / = w «' ;^/ ' '*'■*'<» /■» '* -4. 



• ■« Let r be a positive quantity, in length = f , and let g be inclined to unity at 

 an angle = 6, 6 being positive and less than c, p 



then ^ = r' + 6 + q c . J — \, - ../- 



.:'**♦? .-. r' + 6 -\- qc . J — 1 = m i' + m . a + /j c . J'^^, 



^•. r^=. m V, and &-\rqc^=-m.a.-\-^c 



') 



J 



.'. % is in length = IhST ; 



