By Representation

By Representation refers to a method of distributing property and if defined by New York Estates, Powers, and Trusts Law as:
By representation means a disposition or distribution of property made in the following manner to persons who take as issue of a deceased ancestor:
The property so passing is divided into as many equal shares as there are (i) surviving issue in the generation nearest to the deceased ancestor which contains one or more surviving issue and (ii) deceased issue in the same generation who left surviving issue, if any. Each surviving member in such nearest generation is allocated one share. The remaining shares, if any, are combined and then divided in the same manner among the surviving issue of the deceased issue as if the surviving issue who are allocated a share had predeceased the decedent, without issue.
For example, you left "property in equal shares, by representation, to my children who survive me" and you had three children, two of which died before you. Your property would be broken down into three equal shares and your surviving child would take one third. The other two thirds would be combined and divided into the total number of grandchildren so that each grandchild would take a equal share. For example if one of your deceased children had three children and the other had only one child, each of those grandchildren would take one quarter of the property remaining after your child took the one third share.

This result is very different from a per stirpes distribution. If in the above example, the will read "property in equal shares, per stirpes, to my children who survive me" then your only child grandchild would take one third oof your estate while the three grandchildren would take one ninth of the estate.

Compare: Per Stirpes and Per Capita