NEWTON S PRINCIPIA. 347 



nearly all those cases which it is most interesting to in 

 vestigate. These are : 



&quot; 1. That (B) is approximately an exact differential when 

 the motion is so small that squares and products of u,v,w, 

 and their differential coefficients may be neglected. 



&quot; 2. That (B) is accurately an exact differential at all 

 times when it is so at one instant, and in particular when 

 the motion begins from rest. 



&quot; It has been pointed out by Poisson that the first of 

 these theorems is not true. In fact, the initial motion 

 being arbitrary need not be such as to render (B) an exact 

 differential. 



&quot; Lagrange s proof of the second theorem lies open to 

 some objections.&quot; But it has received two perfectly satis 

 factory demonstrations.* 



Supposing the motion to be such that we may put 



u dx -f v dy + w dz d $ 

 then it easily follows that 



d &amp;lt; d &amp;lt; d d 



~ 

 &amp;gt; 



represent the velocity parallel to axes and along the arc 

 &c. of the curve that the particle in question is describing. 

 At the time I draw curves such that all the particles in 

 them are at that moment moving along tangents to their 

 respective curves. Let s be the arc of any one of these 



curves. The effective accelerating force will be - 2 , and 



the impressed force 



d* dy z dz 



d s d s d s 



* Report to the British Association, 1846, on the progress of Hydrody 

 namics bv Professor Stokes. 



