852 



of smoking is considerably diminished ; likewise the successful ap- 

 plication of naphthalized gas, and of an oil-lamp, to photographic 

 registration. 



The paper concludes with the description of a new method of de- 

 termining the scale and temperature coefficients of the force magne- 

 tometers, by which a greater degree of accuracy is presumed to be 

 attained than by the methods ordinarily employed. Two magnets, 

 designed for self-registering instruments for the observatories at Cam- 

 bridge and Toronto, having been submitted to this method, gave con- 

 sistent results which indicate the law of the temperature coefficient to 

 be sensibly different from that which has hitherto been assumed. 



12. "On certain Properties of the Arithmetical Series whose ulti- 

 mate differences are constant." By Sir Frederick Pollock, Lord 

 Chief-Baron of the Exchequer, F.R.S. &c. 



This paper professes to investigate certain properties of the series 

 of whole numbers whose ultimate differences are constant, and inci- 

 dentally to treat of Fermat's theorem of the polygonal numbers, and 

 some other properties of numbers. 



Its object is to show that the same (or an analogous) property 

 which Fermat discovered in the polygonal numbers belongs to other 

 series of the same order, also to all series of the first order, and pro- 

 bably to all series of all orders. It also proposes to prove the first 

 case of Fermat's theorem (that is of the triangular numbers) from 

 the second case of the squares (which had not before been done), 

 and to dispense with the elaborate proof of Legendre (Theorie des 

 Nombres), finally, to prove all the cases by a method different from 

 that either of Lagrange, Euler, or Legendre. 



It is first shown that an analogous property belongs to all series 

 of the first order (viz. common arithmetical series). The following 

 propositions are then proved as the basis of future reasoning: — 



1. Every triangular number greater than 6 is composed of 3 tri- 

 angular numbers. 



2. Every triangular number greater than 3 is composed of 4 tri- 

 angular numbers. 



3. Any triangular number may be expressed by the form 



4. The sum of any two triangular numbers may be expressed by 

 the same form. 



5. Every number above 7 is the sum of four triangular numbers 

 exactly. 



6. Every number above 29 is the sum of three triangular numbers 

 exactly. 



7. Every multiple of 8 is composed of eight odd squares, and the 

 sum of any 8 odd squares is a multiple of 8. 



8. The following general theorem is then proved : — 

 If p be any odd square, then 



Ap' + ^py + Cp' + Djfy 4- &c. 



will equal 8 odd squares, if 



A + B + C + D + ,&c. 



