Prof. Young oti the Extension o/'Euler's Theorem. 123 



On mixing a hot alcoholic solution of caffein with an alco- 

 holic solution of cyanide of mercury, beautiful needles of a 

 double salt are deposited upon cooling, which correspond 

 most likely to the mercury salt I have just described. A so- 

 lution of caffein in hydrochloric acid gives a beautiful brown 

 precipitate with chloride of palladium ; and the filtered solu- 

 tion deposits another compound in the form of yellow scales, 

 very similar in appearance to iodide of lead. 



CaiFein gives no precipitate with solutions of sulphate of 

 copper, chloride of tin, acetate of lead, and nitrate of suboxide 

 of mercury. When boiled with sesquichloride of iron, a red- 

 dish-brown precipitate subsides upon cooling, which is per- 

 fectly soluble in water, and is most likely a double compound 

 of caffein and sesquichloride of iron. 



XXIII. Note in reference to the exteiision o/Euler's Theorem, 

 By J. R. Young, Professor of Mathematics in Belfast College. 



To Richard Taylor^ Esq. 

 Dear Sir, 



IN the Philosophical Magazine for June last a communica- 

 tion of mine was published respecting an extension of a 

 certain theorem of Euler concerning the products of the sums 

 of squares. At the time that notice was written, I was under 

 the impression that the theorem admitted of an extent of ge- 

 neralization which a further investigation of the matter proves 

 to me has not place. I am now prepared to show that the 

 proposition does not hold beyond the case for eight squares, 

 the formulae for which I have already printed in the Proceed- 

 ings of the Royal Irish Academy ; in the Transactions of 

 which body it is probable that the entire investigation of the 

 theorem for eight squares, and the proof that it does not apply 

 beyond that number, will hereafter appear. 



It may perhaps be interesting to algebraists to find the real 

 limits to this theorem demonstrably established ; and thus to 

 know — in any attempts that may hereafter be made to extend 

 Sir W. R. Hamilton's remarkable and very fertile theory of 

 quaternions — beyond what boundaries such attempts must 

 prove fruitless. 



I remain, dear Sir, 



Very faithfully yours, 



Belfast, July 16, 1847. J. R. YouNG. 



