In ALGORITHM AS58 they explain Euclidian Cluster Analyis.
The article explain the algorithm that the observations are reassigned from cluster k to cluster l if it is nearer the center of latter. That is if $$d^2_l < d^2_k$$ where $d_i$ is the distance from the centre of cluster i. More effective way would be to reassign the observation if the squared deviation from the centre of cluster l is less than that from the centre of cluster k, even if the centres are simultaneously repositioned. That is when: $$ \frac{n_l}{n_l+1}d^2_l < \frac{n_k}{n_k-1}d^2_k $$ where $n_i$ is the number of observations in cluster i.
I need to show how to derive the second formula from the first. Can anybody help me in showing how you can get from the first formula to the second one?
Thank you in advance!