IS^ • DR- James Bottomley on 



The lower of these expressions may be put in the form 



where the letter R denotes a sum of products and powers 

 of small quantities of the third and higher orders ; hence 

 the smaller we make ^x and S/, the smaller will be the error 

 involved in the equation 



^^lxlt=^hhle-'^\\ - e-^^P)^t. (11) 



^-^h YnsLy be written in the form 



— m-r^ox 

 e ax f 



or if we expand 



dq m^/dq^^ 



dq m^(dq\ 

 ax 2 \dxJ 



and I - £-/*^P may be written in the form 

 or if we expand 



1 — e dx 



t9j''-'f&'^^ *•=•' 



therefore (ii) may be written 



^mt = m(i -A^ ^Y^Y^^ - *«•) 



axdt \ dx 2 \dnj J 



\{ we effect the multiplication of the two polynomials oii» 

 the right side, then divide both sides of the equation by 

 ^x^ty and then suppose S;irand S/ to be diminished indefinitely,. 

 we arrive at the following partial differential equation 



fl^^Tchl^f. (13) 



dxdt dx 



