Baron Pollock on some Properties of Numbers, 537 



forms in resisting external pressure. No tubes in use for boilers 

 should ever be made of that form. 



With regard to cylindrical internal flues, the experiments indicate 

 the necessity of an important modification of the ordinary mode of 

 construction, in order to render them secure at tlie high pressures to 

 which they are now almost constantly subjected. If we take a boiler 

 of the ordinary construction, 30 feet long, 7 feet in diameter, and 

 with one or more flues 3 feet diameter, it will be found that the outer 

 shell or envelope is from three to three and a half times as strong in 

 resisting an internal force as the cylindrical flues which have to resist 

 the same external force. This being the case, it is evident that the 

 excess of strength in those parts of the vessel subjected to tension, is 

 actualhj of no use so long as the elements of weakness are present 

 in the other parts subjected to compression. 



To remedy these defects, it is proposed to rivet strong rings of 

 angle iron at intervals along the flue — thus practically reducing its 

 length, or in other words increasing its strength to a uniformity 

 with tliat of tlie exterior shell. This alteration in the existing mode 

 of construction is so simple, and yet so effective, that its adoption 

 may be confidently recommended to the attention of all those in- 

 terested in the construction of vessels so important to the success of 

 our manufacturing system, and yet fraught with such potent elements 

 of disaster when unscientifically constructed or improperly managed. 



"On some Remarkable Relations which obtain among the Roots 

 of the Four Squares into which a Number may be divided, as com- 

 pared with the corresponding Roots of certain other Numbers." By 

 the Rt. Hon. Sir Frederick Pollock, F.R.S., Lord Chief Baron, 



The first property of numbers mentioned in this paper is best il- 

 lustrated by an example — 



132=169 152 = 225. 



These odd numbers may be divided into 4 squares, and the roots may 

 be so arranged that they will have this relation to each other : the 

 middle roots will be the same, and the exterior roots will be, the one 

 2 more, the other 2 less than the corresponding roots of the other. 

 Putting the roots below the number and comparing them, the result 

 is obvious. 



169 225 



0,3,4,12 -2,3,4,14 



-2,4,7,10 -4,4,7,12 



-4,5,8,8 -6,5,8,10 



-6,4,9,6 -8,4,9,8 



Each of the numbers may be divided into 4 squares in 4 different 

 ways with this result, that the two middle roots of each are the same ; 

 and as to the exterior roots they differ by 2, the one being 2 more 

 the other 2 less than the corresponding roots of the other. So 

 comparing 15^ with 17", 



225 289 



4,3,10,10 6,3,10,12 



6,5,10,8 8,5,10,10 



