^fS Prof. Ghallis ofi some Points relating to 



But we have already found, without using equation (9.), that 



F(f)+X(0=^. 

 Hence 



A comparison between these two values of f gives 



The above value of <p must satisfy the following linear equation: 



= a<^ + ^^- + ^V ^ (10) 



It may therefore be assumed that Q{s) and )({t) are linear quan- 

 tities, and accordingly that 



6(5) =5, and ^{t)=ct. 



Also since 4/=0 is the equation of a curve surface, we have 

 in general, 



4'=5; + c' + g'(.r, y,0. 

 Hence 



s=:z + (/ + q[x,i/,t). 



It does not appear possible to satisfy this last condition 

 unless the line of motion be rectilinear ; that is, unless the mo- 

 tion be along an axis, which may be supposed to coincide with 

 the axis oi' z. The function q, being of given form depending 

 on the equation of the surface, cannot express the value of s. 

 This function must therefore disappear on making a; = and 

 «/=0; and we thus obtain 



S=:Z+c\ 



and 



<p=/{z + c'-ct). 



This value applies strictly to motion along the axis of ^. Be- 

 fore proceeding to substitute in equation (10.), it is necessary 

 to express the value of <p for points immediately contiguous to 

 the axis. For this purpose suppose 



f=J{2 + c' + g{x,i/, t)-ct). 



Then substituting the letters^ and q for the functions them- 

 selves, we shall have 



d2f_,dy ^_ d^_ „(d_q__Y.f,^q 



dt ~'' ' dy'^' dz" '-f ' dt^ "-^ \dt ) ^-^ df" ' 



