320 Prof. Matteucci on a new Experimetit in Electricity. 



('tS.) u . w, .7/2 = W3. 



This theorem may be expressed as follows: 



" The continued product of three functions of the form 



is of the same form. ^' .r and ?/ are supposed to be constant, 

 and g, >3 and 6 to be variable. 



In symbols at length, we have 



(44.) {x^q + xyri+yH) {x\q + Ayj,r^+y\^) {wlq-^-Xc^^rt+yl^) 



Xq and j/3 having the values following: 



(45.) x^=xx^Xciq-\-yx^Xc^Yi + {yx^y^+yy^x^-xy^y^)^. 



(46.) y^ -yyiVz S + 'VyxVi 1 + [xy^ Xc^^-x x^y^ -y x^ x^)g. 



This appears to be the most important quadratic modulus 

 theorem which the extended theory of normal couples fur- 

 nishes. It admits of a great number of interesting particular 

 cases. For example, making x in (44.) equal to 1, g = 1, 

 y = Oi the first factor is reduced to 1, and we get the theorem, 



" The product of two functions of the form 

 x^ + xy rj + y'^$ 

 is of the same for jn." 



If, further, we make >) = and Q= 1, we get the well-known 

 theorem, 



" The product of iivo sums of two squares is a sum of two 

 squares.^' 



P.S. " The product of two sums of eight squares is a sum of 

 eight squares." To Euler is due a corresponding theorem 

 relating to sums o^ four squares, which was extended by La- 

 grange, who added coefficients to the squares. As Euler's 

 theorem is connected with Hamilton's quaternions, so my 

 theorem concerning sums of eight squares may be made the 

 basis of a system of octads or sets of eighty and was actually 

 so applied by me about Christmas 1843. I mention this in 

 consequence of the idea suggested by Mr. Cayley in the last 

 Number of this Magazine. But the full statement and proof 

 of the theorem concerning sums of eight squares, and of se- 

 veral other new theorems connected with the doctrine of num- 

 bers, must be reserved for another time. 



XLVII. Account of a new Experiment in Electro-Statical In- 

 duction. By Prof. Matteucci, in a Letter to Dr. Faraday. 

 My dear Faraday, 



I HOPE it will not be disagreeable to you if I describe an 

 experiment which I performed in one of my late lectures 

 on electro-statical induction, in which I dwelt entirely upon 



