On Poncelet's approximate linear Valuation of Surd Forms, 203 



fication of the ammonia. By this temperature Loir and Drion 

 are able to liquefy carbonic acid under the atmospheric pressure. 

 They have also prepared liquid carbonic acid by heating bicar- 

 bonate of soda placed in one of the branches of a sealed tube. 

 On cooling, the carbonate of soda reabsorbs the carbonic acid gas. 



The authors intend to investigate the physical and also 

 chemical properties of the liquid gases prepared at these low 

 temperatures ; under these conditions the ordinary affinities are 

 greatly modified. For instance, 20 cubic centimetres of liquid 

 ammonia placed on a quantity of concentrated sulphuric acid, 

 showed no action at first. Gradually an action was set up and the 

 liquids combined, but with much less violence than might be 

 expected. 



The temperatures were measured in these observations by 

 means of an absolute alcohol thermometer, the fixed points of 

 which were determined by means of the temperature of melting 

 ice, and of that of about 2 pounds of frozen mercury. The 

 temperature of the latter was assumed to be —40° C. 



XXVII. On Poncelet's approximate linear Valuation of Surd 

 Forms. By J. J. Sylvester, A.M., F.R.S., Professor of 

 Mathematics at the Royal Military Academy, Woolwich*. 



M PONCELET' S method of approximately representing 

 • surd forms, and more particularly the square roots of 

 homogeneous quadratic functions by linear functions of the 

 variables, is given in Crelle's Journal, vol. xiii. 1834, pp. 277-291, 

 under the title " Sur la Valeur approchee des radicaux." By 

 this method, as applied to two variables, the resultant of two 

 forces in a plane may be approximately expressed as a linear 

 function of its two components, a case fully considered by M. 

 Poncelet ; and tables have been worked out applicable to this 

 case, which appear to have been found of great utility in some 

 important problems of mechanical and practical engineering. 

 But the illustrious author of this beautiful method has left his 

 theory imperfect in respect of its application to three variables. 



To supply this slight but not unimportant omission, and to 

 indicate how this more general case admits of being treated, more 

 especially with reference to the approximate representation of the 

 resultant of three forces in space as a linear function of its three 

 components, is the object of this communication. At the close 

 of the memoir referred to, M. Poncelet uses these words : — " 11 

 serait inutile de pousser plus loin cet examen (referring to a dis- 



, * Communicated by the Author, having been read at the Mathematical 

 Section of the British Association, June 1860. 



P2 



