528 
Proceedings of the Hoy at Society 
we have 
= S .dqq~ l — 0 . 
Thus, it appears that 
q~ l . dq and its equal — dq~ x . q 
are vectors. 
(2.) From the given equation we have 
da~ • _ i i d<r • i 
jte - um and irr um 
From these 
9 
, d?<r- .du , 0 Tr .dq 1 . du , 0 , 7 . dq~ x 
l - - * — -f 2 uV.i ~ — o=i — -b 2 uV. i « 
J dx dx 
q — i 
dxdy dy 
dy 
From the three equations of this form we obtain by the operations 
Si , S.j , S.&, nine scalar equations, of which the following are 
three : — 
— = 2«S.i %-q , 
dx dy 
*- -2vS.k d JC!q, 
dy dx 
S.f%2q = 
dy M dx 
The last of these, with its two similar equations, shows that 
Si 
.dq 
— i 
q = S.j = 8.k- q - r q = <> 
1 
dx * v dy a dz 
which express Dupin' s theorem for this particular case. 
(3.) If we put for simplicity 
du 
dv 
2 u 
the equations of last section give at once three like 
— q — V.iVv , 
dx 
so that 
and 
dq . q~ x = Y.dpVv , . 
(® (33).) 
Vq - X . q = 2 . iViVv = - 2V?; = - 
or 
V . uq~ l =» 0 
(o (13).) 
