and on Algebra as the Science of Pure Time. 415 
couple (0, 27), but satisfying each the equation (130.), and therefore each entitled to 
be represented by, or included in the meaning of, the general symbol F~'(4,, 42), 
whatever proposed effective couple (4,, ,) may be. For we have only to satisfy, by 
(130.), the two separate equations 
: (136. 
e™ cosa, = by, ee"! sina, = by} ea 
which are equivalent to the three following, 
e N= J bf + b> S77) 
and 
by : bo 
COS a. =~ ——_., SM a = ~—>— 3 (138.) 
J by + bo" VY by + bs 
and if a be the principal solution of these two last equations, we shall have as their 
most general solution 
a,—a+2wn7, (139.) 
while the formula (137.) gives 
a—log../ b+ be: (140.) 
the couple (a, a.) admits therefore of all the following values, consistently with the 
conditions (130.) or (136.), 
(a1) as) =F —" (51; b= Uog J be +b, at+2wn), (141.) 
in which w is any whole number, and a isa number > —7z, but not >, which has its 
cosine and sine respectively equal to the proposed numbers 5,, 5, divided each by the 
square-root of the sum of their squares. ‘To specify any one value of (a,, a,), or 
F—'(5,, 5,), corresponding to any one particular whole number w, we may use the 
symbol r—'(f,, 4.) ; and then the symbol i (41, 42) will denote what may be called 
the principal value of the inverse or logarithmic function-couple r—' (4, 4,), because 
it corresponds to the principal value of the number a, as determined by the condi- 
tions (138.). 
