MATHEMATICAL SECTION. 109 
this by Uf (a, c); elsewhere he uses the notation a‘~) or c(*), which 
is sufficiently distinct for our purpose. 
At the limit we have a(™) a g(*) = e() cos g(”), therefore, 
1 a 
ok shee A od ee ey = bom ight ater e 
dyl=) 1 : 
(=a am) tant (_] + 96 )) (38(@)) 
ie) 
Let e= ‘3 then gy ae aes o" = a. Paki gl) = _](2) and 
1 Dib eras ok J 
im _ly fo ra /= a ea a Fo) ea 
(= Ta tan (J+ am) (39(~)) 
This transformation can be applied also to the more general form: 
Ce 
‘hen f dg ° 40 
= JZ in ¢, 003 ¢, 49) (40) 
0) 
for if, simultaneously to the above algorithm, we express sin ¢, cos ¢, 
A ¢ in terms of sin ¢’, cos ¢’, 4 ¢’, and these again in terms of 
sin yg”, cos g”, A g”, etc., by means of (31), (82), (83), we arrive, 
after a few transformations, at the form 
g(~) 
dg(~) 
~ J eo) cos gi) 
Oo 
I f (>) (sin ¢!™), cos gf@)) (41) 
which is an elementary form if f (sin ¢, cos g, A ¢) is rational 
with respect to sin ¢, cos ¢, A ¢. 
In tracing this process backwards, the quantities may be dis- 
tinguished at the several steps by subprimes, so that we have, at 
the first backward step, 
sin (29 — ¢,) = <t sin 9, = ee sin ¢, (28,) 
é 
a=4(a,+¢);b=3(4,—¢); ¢=Va,¢, (34,) 
Adding, and then also subtracting, sin ¢, from (28,) and dividing 
the difference by the sum, we have the following convenient for- 
mula, also given by Legendre: 
tan (g,—¢) = Stan ? (42) 
