MAJOR P. A. MAOMAHON ON THE COMPOSITIONS OF NUMBERS. 



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Art. 62. To gain familiarity with the new functions I give without proof some of 

 their elementary properties. 



where *, is the sum of the n ih power of the roots. 

 The following expression for a fl - 



a.i"- = f i.-ih*-.+i-<*.-Jin-,+ 

 The result of operations with D f , viz., 



If .<?, denote the sum of the symmetric functions whose partitions contain exactly 

 t parts, we have the companion tables, in which the law is obvious. 



The fundamental properties of these new symmetric functions were communicated 

 by me to Section A of the British Association for the Advancement of Science, at the 

 York meeting, 1906, August 1-8. 



Art. 63. The generating function of N (nbc...) is either 



hoi*... or a (ate y . 



VOL. CCVII. A. P 



