Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

268 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS.
The simplest four-dimensional manifoldnesses in three-dimensional space are that
of all straight lines and that of all spheres. For this reason those contact transformations
between two three-dimensional spaces or which change a three-dimensional space into
itself in such a manner that straight lines are changed into spheres, are the first to
attract attention and have so far been the most fruitful. Lie constructed such a
transformation in his memoir on complexes in the fifth volume of the Iathematische
zAnnalen which has led him to a generalized form* of the theorem of Malus.
Lately this manner of changing straight lines into spheres by contact transformations has been found not to be unique; in fact infinite groups of infinite numbers of
such line-sphere contact transformations have been constructed.
The above observations increase the demand for the resolution of the problem of
determining all continuous groups in four variables. But such contact transformations
need not necessarily be contact transformations of a three-dimensional point space into
itself; for example, if the four variables be interpreted as line-coordinates or spherecoordinates, the corresponding invariant Pfaffians by no means provide that the conditions
for contact transformations of the three-dimensional space into itself be satisfied. It is
precisely because of such a confusion that we find these notions used loosely in a
recent memoirt on the employment of infinitesimal transformations in optics.
* "Lichtstrahlen, die in Pseudonormalensystem bilden, Pseudonormalensystem auf die Pseudokugel des betreffenden
gehen bei jeder Reflexion und Refraction in ein Pseudo- Raumes," Leipziger Berichte, 1896, loc. cit., p. 133.
normalensystem ùber. Sind bei einer solchen Refraction t Hausdorff, "Infinitesimale Abbildungen der Optik,"
die beiden in Betracht kommenden Pseudokugeln (d. h. Leipziger Berichte, 1896, pp. 79-130.
Wellenfiichen) weseltlich verschieden, so bezieht sich jedes