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tangent planes parallel to a given plane have a fixed centre of 
mean position whose position is independent of the direction of the 
given plane. 
2. The centre of mean position of the meeting points of two 
algebraical curves is also the centre of mean position of the meeting 
points of the asymptotes of one of them with the other or with its 
asymptotes. 
3. If through the points of intersection of a curve and a circle 
normals to the curve he drawn, these normals intercept on a trans- 
versal through the centre of the circle segments measured from the 
centre, which are such that the sum of their reciprocals is zero. 
If the circle he drawn to touch the curve at P, and we take for 
the transversal the normal at P, this proposition gives us a con- 
struction for the centre of curvature at P. 
4. Considering all the tangents to a curve parallel to a given line. 
The centre of mean position of the points of contact is the centre 
of mean position of the interventions of the asymptotes. 
The centres of curvature corresponding to the points of contact 
have the same centre of mean position as the points of contact 
themselves ; the sum of all the corresponding radii of curvature is 
zero, and the same is true of the sum of their inverses. 
5. Considering all tangent planes to a surface parallel to a given 
plane, the sum of the principal radii of curvature at the points of 
contact is zero, and the same is true of the sums of their reciprocals. 
The work of the scientific teacher is scarcely less important than 
that of the scientific investigator, although the record of the former 
is more perishable, being at best an oral tradition handed over by 
the immediate disciples of the master. It would appear that in 
this walk Liouville was worthy to rank with his illustrious prede- 
cessor Monge, whose pupils shed such lustre on the French school 
of mathematicians. M. Faye, in his funeral oration, says, “M. 
Liouville was one of the most brilliant professors that ever lectured. 
So lively was my youthful impression of his lectures that to this 
day I have a vivid recollection of the captivating clearness that was 
so peculiarly his own. Accordingly, when in later years I had the 
good fortune to hear him speak at the Institute, I was the less sur- 
prised at the effect which his words produced on my colleagues, who 
marvelled at being able, for a moment, under his guidance, to pene- 
