238 



actually 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 that of the 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. 



II. " On some Remarkable Relations which obtain among the 

 Roots of the Four Squares into which a Number may be 

 divided, as compared with the corresponding Roots of cer- 

 tain other Numbers." By the Rt. Hon. Sir FREDERICK 

 POLLOCK, F.R.S., Lord Chief Baron. Received April 26, 

 1858. 



(Abstract.) 



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

 lustrated by an example 



132=169 15 2 = 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 



