i6o Dr. James Bottomley on 



this equation does not seem to be further integrable. 

 Reverting to (15), assume the following equation 



p = Uo + Uia; + Uaa;^ + Uga;^ + U^a;* + <fec. ; (20) 



wherein the coefficients Uo, Ui, U2 &c., may be functions of t. 

 Differentiating with respect to x we obtain 



^ = Ui + 2U2itr + SUa^r^ + ^\}^x^ + SUgO?* + &c. ; (21) 



denoting differentiation with respect to t by using accented 

 letters, from the last equation we obtain 



^ = U'l + 2U> + SU'gO^ + 4U',a;3 + SU'bo?* + &c. (22) 



Equation (15) may be written in the form 



Cul) 0.13 



'°Sdw«-'°s;&=>°8(-«) + <:p-*^; (23) 



substituting from (20), (21), and (22) in the last equation we 

 obtain 



log(U'i + 2UV + SU'gor^ + &c.) - log(Ui + TC^x + SUs*^ + &c.) 

 = log( - a) + cUo + a;(cUi - 6) + cV^.x'^ + cV^a? + cU^a;* - &c. (24) 



Expand log(Ui+2U2^+3U3;r2+&c.) in a series of powers 

 cfjir; put V for 2Ua^+3U3;ir2 + &c. ; then when ;ir vanishes 

 V also vanishes, hence the first term in the expansion of 

 log(Ui + V) will be logUi ; the coefficient of ;ir will be 



/^L ^_I\ and of .^ H^(-^^ ^-^U 

 tUi + V dx],J ^^"^ ""^ ^ ' 2\dx\Vi + V dx)],J 



and generally the coefficient of ;ir^+^ will be 



assume 



v, ^ = Ao + Ai^; + A2^^ + Agic' + Ai^;* + <fec. ; (25) 



if Aji be the coefficient of x'^ in this expansion we shall 

 have 



From the equation 



V = 2U2a; + 3U3^2 + 4:ViX^ + 5V,og* + &c., (27) 



