On a Theorem relating to the Products of Sums of Squares. 515 



When the positive terminal is coated and the negative one bare, the dark 

 space appears on the point of the bare wii-e, the wire itself being surrounded 

 by a lambent ilame ; but with the converse arrangement there is no such 

 dark space. All this is much in favour of interference taking place, the 

 coincidence of positive and negative phases of the discharge producing at 

 certain points mutual neutralization. 



Description of Plate. 

 Plate V. 

 Figures 1 to 10 show the spots and rings in the order referred to : it 

 should be observed that printed figm'es give but a very imperfect 

 notion of the actual eifects. 

 Fig. 11 is the coil apparatus, the contact breaker being in front. 

 Fig. 12. The air-pump, of a construction which I proposed many years ago, 

 and have found most useful for electrical or chemical experiments 

 on gases. 



P. An inperf orate piston, with a conical end, which, when pressed 

 down, fits accurately the end of the tube, the apex touching the 

 valve V, which opens outwards. 



A. Aperture for the air to rush from the receiver when the piston 

 has been drawn beyond it. 



B. Bladder containing the gas to be experimented on. 



The piston-rod works air-tight in a collar of leathers, and the 

 operation of the pump will be easily understood without further 

 description. 



If it be requii-ed to examine the gas after experiment, a bladder, 

 or tube leading.to a pneumatic trough, can be attached at the ex- 

 tremity over the valve V. 



LXXX. Demonstration of a Theorem relating to the Products 

 of Sums of Squares. By Arthur Cayley*. 



MR. KIRKMAN, in his paper '' On Pluquaternions and 

 Homoid Products of Sums of n Squares" (Phil. Mag. 

 S. 3. vol. xxxiii. p. 447), quotes from a note of mine the followiug 

 passage : — " The complete test of the possibility of the product 

 of 2" squares by 2" squares reducing itself to a sum of 2" squares 

 is the following : forming the complete systems of triplets for 

 (2" — 1) things, if eab, ecd, fac, fdb be any four of them, we 

 must have, paying attention to the signs alone, 

 ( ± eab) ( ± eci) = ( ±fac) ( ±fdb) ; 

 i. e. if the first two are of the same sign, the last two must be 

 so also, and vice versd ; I believe that, for a system of seven, 

 two conditions of this kind being satisfied would imply the satis- 

 faction of all the others : it remains to be shown that the com- 

 plete system of conditions cannot be satisfied for fifteen things." 

 I propose to explain the meaning of the theorem, and to establish 

 the truth of it, without in any way assuming the existence of 

 imaginary units. 



* Coiumuuicated by the Author. 

 2L2 



