1 66 Dr. James Bottomley on 



if in this equation we substitute differential coefficients 

 with respect to / for the accented letters Ug^ \5}, Ui\ and 

 integrate we shall obtain the equation 



/I /-/^logUi J \dt) W ,, \ ,, >| ,, ., 





We may write 





\ dt 



) u..\ . / 



1-1 V / '2 



^USA'"u? 



+ 



d^ Ui U^Ui Uir 



in the form 2 



by means of (39) and (41) this may be put in the form 



(cUi - 6) (cUi - 61ogUi + 2r2 + ^^^ i 

 hence equations (43) may be transformed into 



Hence U3 has been transformed into an integrable function 



of Ui ; by a continuation of the process U4, Us, &c., may be 



obtained as functions of Ui, and the method seems generally 



applicable ; for if we equate the coefficient .of x^^^ in the 



expansion of 



1 a._^ ^\ it 

 ^dxdt ^dx 



with the coefficient of the same power in the expansion 



cUo + (cUi - h)x + cJJ^x^ + cV^x^ + CU4X* + &c., 



we shall obtain an equation which may be put in the form, 



(?i + 2)r —jfY — T^} "*" f^iJ^ction of 



(UVo ^\, .... W, Ux^ ; U,+„ U,, . . . U2, Ui) =cU„+, ; 

 this equation may for brevity be written 



