188 Prof. Sylvester on the Motion and Rest of rigid Bodies. 



the gaseous state without any condensation of volume, it may 

 be presumed that if we subtract the refractive index of hydro- 

 cyanic acid from that of cyanogen, viz. -275 from -316 = •041 

 for the refractive index of liquid hydrogen. 

 82 Blackfriars Road, London, Feb. 1, 1836. 



XXXV. On the Motion and Best of rigid Bodies. By 

 J. J. Sylvester, Professor of Natural Philosojihy in Uni- 

 versity College, London.* 



IN the subjoined investigation, which, as far as I know, is 

 my own, I apply the same method to rigid as in a pre- 

 ceding paper I applied to fluid systems. 



Let X y z be the coordinates of any particle in a rigid 

 body ; 



.r' y %' the coordinates of some other particle, and 

 let x = X -\- h y' = y -\- Ic z' =■ z -\- I. 



Call A i2', A^, A-t: the increments which x, y, z receive 

 after the lapse of a small interval of time; so that terms in 

 which they enter in two or more dimensions may be neg- 

 lected. 



rr., . , ,N . d /S. X ^ dAX , dX T , T, 



Then A (x') = Ax+ —, — h f -^ — k+ -^— .1 + V 

 ^ ' ax dy dz 



, „ . . rZ A ?/ - d Ay J dAy, r\ 

 A{y')=Ay^.^J.+ -^.k^^M+Q 



, ,. dAz , dAz , dAz , ^ 



P, Q, R containing binary and higher combinations of h, h, I, 

 which we shall have no occasion to express. 



At the commencement of the interval the squared distance 

 of the two particles was {x'—xY+ (y' -yY + {z' — z)^; at the 

 end of the interval the distance squared is 



{x'-x+ A{'x)-AxY+ {y'-y + A{y')-Ayf+{z'~z 

 ^A{z')-Azy. 

 and these two expressions must be the same by the conditions 

 of rigidity whatever //, Ic, and I may be; i.e. 



d A X d Ay dA. 



h^^+p+r- = (^ 



+ 



d; 



f , d Az , dAZ, dA 



for all values of h, Ic, and I. 



* Comniuniealed by the Author. 



