APPLICABLE TO THE MOTION OF FLUIDS. 385 



Since u = N-~, by equation (13), 



dx' dt + \dx* + df + dx 2 ) ~ "' 



whence by differentiating with respect to x, 



d*cj> d± d*<f> dj> d<j> d<f> (d$ d*$ d$ d 2 (j> d<p d 2 <p \ 



du _ da?' dt dxdt' dx dt * dx\dx' da? dy' dxdy dx'dxdxl 

 dx~ d(fi dfi e^F" + M d(j> 2 d(j> 2 \* ' 



da? + df + dx? \dx 2 + dy 2 + dx?) 



Similar expressions having been obtained for -=- and —r- , it will 



be found by adding them together, and having regard to the formula 



in Art. 2, for — I- — , that, 

 r r 



/du dv dw\/d<p : d<p* d(p s \_d 2 cp d<p d 2 <p dip d 2 (p d<p d 2 <f> d<\> 

 \dx dy dx)\dx 2 dy 2 dx 2 ) dx 2 dt dxdt' dx dy" ' dt dydt' dy 



d 2 (p d$ d?cj> dj> dcfr (d<j? d<p djjt\ * / 1 1_\ 



+ d¥'~dl ~ dx~dt' dx* dt \da? + dy 2 + dx 2 ) \r + 7) ' 



which equation may be reduced as follows to one of a simpler form. 

 Equation (13) gives, 



<!£ _. %*(*<£ .tip + *£y - v* & -*- *£ 4. *£) 



dt 2 \dx 2 df dx 2 ) ' \dx? T df dx 2 ) ' 



Hence d<p* ^ d£ = ± d£ 



wence ' dx* + dy 2 dx 2 V 2 ' df 



Also, by multiplying equation (13) by N-^- , it will appear that, 



dt dx 



and this equation by differentiating with respect to x gives, 



