In the last lecture we Considered for two dimensions the problem of making up a complete list (i) of all orthogonally inequivalent finite groups of homogeneous orthogonal transformations, (ii) of all such groups as have invariant lattices, (iii) of all unimodularly inequivalent finite groups of homogenous transformations with integral coefficients, (iv) of all unimodularly inequivalent discontinuous groups of non-homogeneous linear transformations which contain the translations with integral coordinates but no other translations.

Problem (i) was answered by Leonardo’s list

*C _{n}*,

(ii) by limiting the index*n*in it to the values