142 SCIENCE PROGRESS 



There is little doubt that the theorem is correct, and that there is some 

 way of deducing it from theta-function series ; but it was not proved in 

 1916, and I have seen no proof myself.^ Not improbably, some graphical 

 method will provide the key to the puzzle. 



Major MacMahon gives other instances of conjectural theorems, and in 

 this way the work is certain to stimulate research. In the matter of magic 

 squares, too, he has a very interesting and valuable chapter. In fact, he 

 does obtain a " crude " enumerating function, but the reduction of this 

 to the true form, even for small values of n, seems hopelessly laborious. 



Enough, I hope, has been said to give some idea of the nature and extent 

 of Major MacMahon's achievements. One great advantage which the 

 Combinatory Analysis has for a student is that he can start reading it with 

 a very small amount of mathematical knowledge ; and everything possible 

 is done for him by way of elucidation. This is not to say that the book 

 can be read as you run ; the subject is too difficult for that ; but hardly 

 anyone of capacity can fail to be fascinated by the subject-matter, and by 

 the way in which it is presented. 



1 Since the above was written I have seen the obituary notice of Ramanu- 

 jan in Proc. L.M.S. (May 1921). From this it appears that several of his 

 theorems (probably including the above) have been recently proved by Prof. 

 Rogers and Messrs. H. B. C. Darling and L. J. Mordell. 



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