Prof. Sylvester on the Motion and Rest of rigid, Bodies. 1 89 



Hence rejecting infinitesimals of the second order and 

 equating to zero separately the coefficients of A-, F, P., and of 

 k /, I k, h k, we have 



d Ax , . d Ajj d Az ^ ,j. 



dx ^ ' dz dy 



By differentiating (rZ), (e), (/) with respect to x^ x, y re- 

 spectively, and substituting from (a), (6), (c), we obtain 

 d^Ay d^Az ^ d~ A x 



a;:- dx^ dy'' 



By differentiating the same with respect to ?/, s, jr respect- 

 ively, and proceeding as before, we have 



drAz d^Ax ^ d?Az 



dy dz- dx' 



Thus, then, we have 



dAx ^ d^Ax ^ d-Ax 

 dx dy- dz-^ 



dAy ^ ^ d^Ay ^ ^ ^^ = o 



rfAj; (Z'^A^r ^ d' Az 



— T— = —-^=0 -—- = 

 a ;: dx' dy- 



.\ Ax = K + By + Cz (o.) 



A3/ = D + E~ + r.2? {p.) 



Az=G ^nx + Ky {q.) 



A, B, C, D, E, F, being constant for a given instant of time ; 

 between which by virtue of the = "^ {d), (e), {/), we have 

 the I'elations 



E + K=o H + C=:0 B+F=0 

 If we call ti, V, to the three component velocities of the 

 particles at x, j/, z parallel to the three axes, and X^ Y^ Z^ 

 the three internal forces, it is at once seen that ?/, v, iio, as 

 also A X^ Y^ A, Z; must be subject to the same equations as 

 limit Ax, Ay, Az; 



so that ti = a + yy — ^z {\) A X, = «, + 7j3/ — B,z [h.) 

 v=h + uz — yx{2) AY, = b, + a,z—yx, {/.) 

 to = c + /3.r - ay (3) A Z^ = c, + ^,x-u,y {k,) 



