132 



magnet and the electric currents evolved in them carried off to the 

 galvanometer. Now, Vv'hether the discs be of paramagnetic or dia- 

 magnetic metals, whether of iron, or bismuth, or copper, or tin, or 

 lead, the direction of the current produced shows, that the lines of 

 magnetic force passing through the metals is the same in all the 

 cases, and hence ihe polarity within them the same. 



The author then gives a more explicit meaning, in accordance 

 with the definition of line of magnetic force contained in this paper, 

 to some of the expressions used in the three last series of his 

 Researches on Magnetic Condition, Atmospheric Magnetism, &c. : 

 and by referring to former results obtained since the year 1830, 

 illustrates how much the idea of lines of force has influenced the 

 course of his investigations, and the results obtained at different 

 times, and the extent to which he has been indebted to it ; and then, 

 recommending for many special reasons the mode of examining 

 magnetic forces by the aid of a moving conductor, he brings for the 

 present his subject to a conclusion. 



December 18, 1851. 

 WILLIAM SPENCE, Esq., V.P., in the Chair. 

 The following papers were read : — 



" A Proof (by means of a series) that every Number is composed 

 of 4 Square Numbers, or less, without reference to the properties of 

 Prime Numbers." By Sir Frederick Pollock, Lord Chief Baron, 

 F.R.S. &c. Received December 18. 



The paper contains a proof, that if every number of the form 

 8« + 4 is composed of 4 odd squares, then every number whatever 

 must be composed of 4 square numbers or less ; also a proof of the 

 converse of this, viz. that if every number is composed of 4 square 

 numbers or less, then every number of the form 8?2-{-4 must be com- 

 posed of 4 odd squares. 



It is then proposed to show that every number of the form 8^^^-4 

 is composed of 4 odd squares, by taking a number of the form 8?^^- 4, 

 viz. an odd square +3. and showing that 8?z + 4 in that case is 

 divisible into 4 odd squares (other than the odd square and 1, 1, 1); 

 thus \Qn-+^n+\ is a form that includes every odd square, and 

 16/2-+8«-f-4 is divisible into 



4yz2 + 4?i+l, 



4^2 + 4/7+1, 



4w- + 4w-fl, 



8 is then added, and the sum is shown to be still divisible into 4 odd 

 squares ; and again 8, and so on, until by successive additions of 

 8 + 8 + 8, &c., the quantity added to IQn'^-j^Sn becomes equal to the 

 original term with which the operation commenced. The odd 

 squares +3 form the series 4, 12, 28, 52, 84, &c. ; and if the sue- 



