As we proceed to higher and higher orders of aberrations beyond the seventh, the compass of the iteration equations as hitherto set out gets rapidly out of hand. This is partly due to a certain wastefulness inherent in not explicitly retaining certain combinations of coefficients that enter into the equations of order 2n+1, because the same combinations recur in the equations of all orders exceeding 2n+1. An appropriate notation is introduced, to take this feature into account. The revised iteration equations of orders five and seven are written down, though the advantages of proceeding as here suggested do not make themselves fully felt until we go at least to the ninth order. A modification of the iteration equations of the highest order to be contemplated in a given problem is considered, thus leading to further computational economy.

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