569 
NOTES AND BEFEBENCES. 
518. Ribaucour, C. R., t. lxxy. (1872), pp. 533—536, referring to my Note remarks 
that the condition can be (by means of the imaginary coordinates of M. Ossian Bonnet) 
expressed in a simple form communicated by him to the Philomathic Society, May, 
1870. I reproduce this investigation, although it is not easy to present it in a quite 
intelligible form. We take p = f(x, y, z) to represent a family of surfaces belonging to 
a triply orthotomic system, and consider two neighbouring surfaces (A) and (A') corre 
sponding to the values z and z + dz; A and A' the two points where they meet the 
trajectories of the surfaces; AT, A'T' the tangents to the curves of curvature of the 
same system at A, A' respectively. Then according to the remark of M. Levy, it is 
to be expressed that these lines meet, and this is done by expressing that along the 
trajectory A A', the variation of the angle of AT with the osculating plane at A is 
equal to the angle of the osculating planes at A, A' respectively. 
Let B' be the spherical image of A', the plane OBB' is parallel to the osculating 
plane at A of the trajectory, and the angle of the two osculating planes measures 
the geodesic curvature of BB': denote this by d<y. 
Let /3 be the angle of BB' with BX, 6 the angle of AT with BX, /3 — 6 is the 
angle of AT with the osculating plane at A of the trajectory: d/3 — cW = dy. Introducing 
the symmetric imaginary coordinates x and y, we write 
a = 
dp 
X 2 dx ’ 
b = 
dp 
X 2 dy 
1 d'P 7 , .^ n da db 1 7 
c =— j—j- , ds- = 4\ 2 -y -j- dx dy. 
X- dxdy dx dy ° 
But dx and dy being the increments of x, y corresponding to dz in the passage 
from A to A', then by a theorem of M. Liouville 
the condition thus is 
C. VIII. 
<h= d e- i {iaL dx -\Ty dy )' 
dd = i 
dX -, dX \ 
dx -\T y dy )' 
X dx 
72