20 



actly in the same way as this latter involves that for four : and, 

 to be correct and general, it must yield the eight-square form 

 when, by the suppression of four of the binomials in any row, 

 the sixteen squares are actually reduced to eight. 



Now it is shewn that these conditions cannot be accom- 

 plished ; unless, indeed, under special and peculiar limitations, 

 which are pointed out. In proceeding to construct the six- 

 teen-square form, providing, as we go on, for the demands of 

 that for eight, since these are necessarily implied in those for 

 sixteen, we find our progress, beyond a certain stage, to be 

 impossible ; inasmuch as a step further imperatively requires 

 that a preceding step should be modified, which modification 

 is fatal to the accuracy of the portion already constructed. It 

 is hence concluded that the eight-square formula cannot be a 

 particular case of one more general for sixteen, but is itself 

 the most advanced modular theorem that exists. 



Towards the close of the paper the author enters upon 

 some collateral investigations concerning squares and products ; 

 and, among other things, offers a short method of establishing, 

 without imaginaries, a very beautiful triplet theorem disco- 

 vered by John T.Graves, Esq., and printed, with its investi- 

 tion, in the Philosophical Magazine for 1845. 



December 13th, 1847. 

 JOHN ANSTER, LL.D., in the Chair. 

 The following letter from Mr. Staunton was read : 



" Longhridge House, near Warwick, 



''Dec.Ath, 1847. 



" Gentlemen, — I have the pleasure of receiving, this 



morning, the Fyshers Irchington matrix, and lose no time in 



acknowledging its receipt, as I do with additional satisfaction, 



