and on Co-resolvents. 177 



for convenience only. In the abridged notation (a) will be 

 wi'itten 



y"' + ^Ty + sy = - - - (h) 



4. Change the dependent variable fi'om y to Y where y 

 and Y are connected by the relation 



y = uY - - - -(c) 



proceeding as follows : — 



y"' = (u Y)"' = u T" + 3 ii' Y' + 3 u' F + vJ" F, 



3 r 2/' = 3 r (^^ F)' = Zr%iY'-\-Zrv!Y 



sy = s (u Y) = suY 



whence, bearing in mind the relation (6), 



uY'"-{-S ii' Y" + 3 {%f + r II) Y' + (u'" + 3 r ^6' + s iC) 7= - (cQ 



or, dividing by ii, 



F-+ si^LF^ + SpJl' + J Y'^{~ ^-'^T~-\-^\ F=0 - (e) 



u \\i ) \ %i u J 



5. Next change the independent variable from o^ to ^ and 



fd t\ ^ 

 we find, after dividing by I -y— I or (ty, that (e) becomes, 



if we confine, for a moment, the change to F', Y'\ and Y'/' 



where F (u, x, t) is a function which it is not at present 

 necessary to develope. 



6. Hence, one of the conditions of the proposed transfor- 

 mation being that the second term of (/) should disappear, 

 or, what amounts to the sa,me thing, that the co-efficient of 

 Y" should vanish, we have, first, 



— . X V — - - - - iq) 



10 x^ ^^^ 



which reduces to 



^-%=0 (/.) 



