MATHEMATICAL SECTION. 111 
To discover the relations for the first step we have to determine 
¢, from the equations 
Wn =1@)n = ¢y (46) 
Place in (12) wu =(¢,)y,, then 3 u = (¢,)y, and u_y, = ¢,; (5) -/1 
= ¢,, and consequently 
sin 7¢, = 1 — cos ¢, +1+A¢, 
cos*y, =A¢g,-+ cos ¢¥, +1l+Ag, 
4%, =f) +49,+y77cosg,+1+4¢9, (47) 
From (32) and (33) we derive with due regard to (43) 
a,ay,=3(aa¢ +z) 
b 
g4¢,=t(aae— x2) (48) 
and eliminating ¢g, from (47) by means of (48) there result the 
relations 
’ a 1—Ag 
Se i Gin Taale 
gyi it Senere i, 
me fe 6 (Lag) 
awtg+t+b 
2 — 
ath =a £49) oS) 
whence also 
» ____ asin gy 
ae a, + ¢, sin *¢, 
__ a, cos $, Ag, 
Ne tan feed 
— } 2 
~ a, + ¢ sin *¢, 
This is Gauss’ transformation. For practical use it is far less 
convenient than that given above. 
Instead of (46) we might have assumed — 
m (,)y, = 2 (¢,)y, = 2n <1. gy (mand n integers) (51) 
For any special values of m and n we can express, by means of 
