PAPER BY PROF. HELMHOLTZ. 35 



If the conditions (Ic) are satisfied at the point whose coordinates 

 are i;, i), ^, and if we designate the values of w, v, w, and their dif- 

 ferential quotients as follows : 



We obtain for the point whose coordinates x, y, z, differ differentially 

 from i;, l), ] : 



u=A + a {x-x) + y {y-\))-\-p {z->>), 



v = B-^y{x-v.) + h {y-\)) + a{z-^), 



iv=C+f3 {x-v) + a{y-\)) + c (c-^), 

 or when we i)ut : 



(p=A {x-v) ^B {y-\)) +C{z-]) 



+ h a{x-xX-+hh[y-\)f+hc{z-sf 



+ cx{y-\)) {z-s) +f5 (^-i;) {z-s) +k (^-iO (y-\)), 



there results : 



^ q) i) qj _ J tp 



~ :) x^ ~ ,^y^ ~ ;) z 



It is well known that by a proper selection of another system of rec- 

 tangular coordinates x^, y^ Zi^ whose origin is at the point i;, t), j, the ex- 

 pression for q) can be brought into the form : 



q)=Ai Xi-\-Bi yi+Ci Zi-\-^ «i a?i^+i&i 2/1^+^ Ci ^1^ 



where the component velocities ?ti, Vi, Wi^ along these new coordinate 

 axes have the values : 



u^=Ai-\-aiXi, ri = Bi-\-bi yu iCi = Ci-\-Ci Zi. 



The velocity Ui parallel to the axis of Xi is therefore alike for all 

 liquid particles that have the same value of Xi. therefore particles that 

 at the beginning of the elementary time dt lie in a plane parallel to 

 that of yi Zi are also still in that plane at the end of the elementary time 

 dt. This same proposition is true for the planes Xi yx and Xi Zi. There- 

 fore when we imagine a parallelopipedon bounded by three planes 

 parallel to the last named coordinate planes and infinitely near to 

 them, the liquid particles inclosed therein still form at the end of the 

 time dt a rectangular parallelopipedon whose surfaces are parallel to 

 the same coordinate planes. Therefore the whole motion of such an 

 indefinitely small parallelopipedon is, under the assumption expressed 



