89 
( .cie .d&\- , t m .doy , 
Vdz ~dy) + Vdx X dz) + 
0 2 
where 0 is a function of xyz , which satisfies the equation 
Liouville gives this theorem in terms of generalised coordinates 
for any system of particles, and points out that it opens up a new 
method of treatment leading readily to all the known results of 
Theoretical Dynamics. 
During the latter part of his life, Liouville’s researches were 
almost entirely directed to the Theory of Numbers. From 1857 to 
1873 we have a list of over 200 notes and memoirs on this subject, 
all published in his own journal. A few occur with earlier dates, 
for examples the following : — 
“ On the equation Z 2n — Y 2n = 2X W .” Jour, do Math., v., 1840. 
“ On a Theorem of the Indeterminate Analysis.” Comptes Rendus , 
x., 1840. 
“ On the Two Forms x 2 + y 2 + z 2 + t 2 , x 2 + 2 y 2 + 3 z 2 -f- 6t 2 .” Jour, 
de Math., x., 1845. 
The most important of all the memoirs on this subject are the 
series entitled “On some General Formuke which may be useful in 
the Theory of Numbers.” Jour, de Math., vols. iii.-viii., New 
Ser., 1858-1863. 
Yery few of the longer memoirs are devoted to Pure Geometry ; 
but many interesting and novel geometrical theorems occur in- 
cidentally in Liouville’s mathematical writings. A full account of 
these will be found in the third chapter of Chasles’ “ Eeport on the 
Progress of Geometry in France.” — Recueil de Rapports sur VRtat 
des Retires et les Pr ogres des Sciences en France, Paris, 1870. 
We may mention here some of the results arrived at in two 
memoirs, “ On certain general Geometrical Propositions, and on 
the Theory of Elimination in Algebraical Equations (Jour. deMath., 
vi., 1841), and “Developments of a Geometrical Theorem” {Jour, 
de Math., 1844). The following results among others are arrived 
at : — 
1. The points of contact of a geometrical surface with all the 
