PROF. A. C. PIXoN oN S'iTK.M LIOUVILLE HAKMOMC KXI'ANsIoVs. 421 



itii;- cn,,s/n,,/. nr< nil of the xnin<- ..,/,-, ,,(',,,,!</, t ml, <i* exp a (/> '<), 

 be f loii-fi- .!//./ ///,/c.vx r-///.v /-. :/-,!. 



14. Hie values of u (6, a), !"(/>.'<)... can be written down M follows : 



L'-l tin' succi->M\e intervals into which /< ^ is divided Ix- denoted by #,, rt,, ..., ft,, 

 and tin- valurs of / in those interval-, respectively liy /'i, /';,, ..., /'. : also let ,\ = /*. 

 Then 



tt(6,o).^to^Hsfid2fflL^i^^ 



f23 . e 3 )'3 ... f.-i?',., 



where *,, e 2 ..... e,, are all 1 and 2 refers to the 2" ways of taking them. The 

 product in the numerator of the fractional factor contains it I hinoiiiial factors. 



exp 



" exp 



|/i ... f,, 



It may, in fact, lx- veritieil that 



!>< dU C 2 



~ 



so that , U satisfy the diffen-ntial e(|uations assigned ; also, when n = the 

 expressions reduce to the corresponding ones for u 1 intervals, and therefore u, U 

 are continuous throughout as functions of 1, ; lastly, at the beginning of the second 

 interval 



u = j sinh /#,, U = cosh /#,. 



This completes the verification for , U, and V, / may l>e treated similarly. 



The approximations generally used in the treatment- of the equation (l) might lie 

 derived from these by neglecting all the terms except those in which e, = t, = ... = f ., 

 that is, the terms which contain one or more of the differences r, r 3 , > r n , ... as 

 factors, and in the two terms which are left putting Sx/rl/vTi lor <, + >+!. Thus 

 (l>, a), for instance, would reduce to 



\/>\r n sinh I (h a). 



This simplified form is clearly not admissible unless all the differences /, /,. /,.-/.-,, ... 

 are small. 



