386 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



d*<p d(j> _ d 2 <p d±_ }_ d<p* {2u dV du 

 ~dx* ' It ~ dxdt ' ~dx ~ V % ' dt* \ V '~dx dx 



d*<p d<p dty d<j> 1_ djff^ dV _dt\ 



S ° dtf ' dt dydt' dy V*' dt\ V ' dy dy)' 



d*<p d<p d*<p d<j> _ J_ dcj?(2w_ dV _dw\ 

 d% 2 dt~ ~iRdi'd%~V*' df\V dz d%)' 



When the several values thus obtained are substituted in the fore- 

 going equation, the result is, 



du dv dw _ u dV v dV w dV i\ 1\ 



dx dy d% ~~ V dx V dy V d% \r r) ' 



If now the condition be introduced that the variation from one point 

 to another of space be in the line of motion, we shall have, 



u dx v _ dy w _d% 



r'di* T m d*' r~~ds'' 



and the above result is reduced to the following, 



du dv dw _dV rrf^- 1\ 

 dx dy d% ds \r r'l ' 



Consequently by reasoning exactly as in Art. 4, an equation the 

 same as (5) results: and by eliminating p from this equation by means 

 of equation (14), the equation (10) is reproduced. We may therefore 

 conclude that the same differential equation of the second order, in ivhich 

 V is the principal variable, applies to curvilinear as to rectilinear motion, 

 the variation of the co-ordinates at a given time being from one point to 

 another in the line of motion. 



14. The reason of this result will be seen by the following consi- 

 derations. Conceive two surfaces of displacement to be drawn at a given 

 instant indefinitely near each other, one of which passes through the 

 point P given in position. On this surface describe an indefinitely small 

 rectangular area having P at its centre, and having its sides in planes 

 of greatest and least curvature. On the other surface take a similar 



