411 On the general Differential Equations of Hydrodynamics. 



motion may be modified in any manner by disturbing causes. 

 But as the foregoing argument has been conducted prior to the 

 consideration of particular disturbances, the inference to be drawn 

 from the result is, that the intcgrability of udx + vdy + wdx is the 

 analytical exponent of rectilinear motion which takes place in the 

 fluid by the mutual action of its parts. The inference so drawn 

 is in accordance with the third remark in art. 9. It will be seen 

 hereafter that this general inference explains the rectilinear trans- 

 mission of light. 



12. It will be proper here to meet an objection which possibly 

 may be raised against the foregoing reasoning. If we suppose 

 udx + vdy-\-wdz to be integrable of itself, and substitute (d<j>) for 

 it, by applying to (d<j>) the same process that was applied to 

 (dyjr) we should obtain the equation 



#,#!,#! , # 2 _ 



dt \&« ^ dy* "*" dz* ~ u ' 

 which is manifestly untrue. The answer to this objection is, 

 that ty is a function altogether different from </>, by reason of 

 the function X, which cannot be left out of account even when 

 udx + vdy + wdz is integrable without a factor. This may be 

 proved as follows. Since by supposition \(cfyr) = (^<£), we may 



put ., '*£j' for X, so that <£ = ^(-^, t). Hence 



# __ <?- XW' *) : d X -a# & c# 



doc " dyjr dx dx 3 

 But at the same time 



dt dty dt dt dt dt ' 



the second term being the partial differential coefficient of %(^, t) 

 with respect to t. Consequently, by substituting in (3), the 

 result is 



4 d<p d<j>* dt* _ d. x w,t) 



dt + dx* "*" dy 2 "*" dz 2 dt 3 



which equation, strictly deduced from (3) on the supposition that 

 udx-\-vdy-\-wdz is integrable, is different from that obtained 

 above on the same supposition. This shows that it is not legi- 

 timate to apply to (d<p) the same reasoning as to (dyjr). 



13. Prom the last equation an inference of some importance 

 may be drawn. There are cases of motion, as will appear here- 

 after, in which udx + vdy + wdz is approximately an exact differ- 

 ential for the whole of the fluid and during the whole of the 

 motion, and in the treatment of which the square of the velocity 



is neglected in comparison with ~, If this be done in the above 



