cl£l ^0 
where O is considered as a function of r, v t , 0, p, q. The symbols as 
~ CtlJ CL Cj 
employed by Hansen, mean that the differentiations are to be performed as if 0 was 
a function of p, q, such that 
¿9 - d& d „ | cl ® da - 1 d P-P d< l ■ 
dp P dq ^ cos i (1 + cos i) ’ 
the last two equations being therefore nothing else than what Hansen represents by 
7 na cos 2 i dVl 7 
= dj dt ’ 
7 na cos 2 i dil 7 
dq = W^) ~dp 
X, the departure, r for t, 
i.e. X is what v, becomes when t, in so far as it enters explicitly, and not through 
the variable elements, is replaced by the new variable r; so that, in fact X = 
The values of r, v t could be at once found from those of p, X by changing t into t; 
and to determine the values of p, X, Hansen proceeds as follows: 
writing 
X = n (£ t), 
Ip = T (£, t) + (3, 
where £ and ¡3 are new variables functions of t and t, and II, T are arbitrary 
functional symbols; so that if z, w are what f, ¡3 become when t is changed into t, 
we should have 
v, = n (z, t), 
lr=F (z, t) + w; 
then the foregoing equations give 
dip -pv / y dip d(3 
dr ~ ^ ’ dr + dr’ 
t- r ' ( £0t+r (( f,0 + f; 
[where the accents and strokes denote differentiation in regard to £, t respectively].