115 



an approximate integral ; namely, by computing from (C") the ap- 

 proximate expressions of 



and trying whether their sum is unity, when rf is neglected. Putting 

 for this purpose the expression (C") under the form 



V K fl" 1^ 



— = « — riO) + 2(2 + A) ~ 8(2 + Af ^ 16(2 + Af 



(A — ^wy (32M>,-1 6«>6--3^y (^ — 8m>JV „„ 



"♦* 8(2 + Ar ^ 16(2+^)6 "^ 8(2+^)'' ' ^ ' 



we find, by differentiation, 

 1 dv 



+ 



3«'' 



5»' 



f/- «f2 2(2 + Af ^ 8(2 + ^)* 16(2 + Af 



(^— 8m>,V- _ 3(32«>, — 16tOe— 3^y _ 1{A - 8k>JV 

 2(2 +^)'^ 8(2 + J)'' 8(2+^)^ ' 



2 «/K 1 



3,^ 



, (^ -8u>J, 



^ rf« 2 -f- J 2(2 + ^)3 "T- 8(2 + ^f "•' 2(2 + Af 



3(32w, — 16W, — 3J)„' 3(J — 8«;JV 

 ■*" 8(2 + Af + 4(2 + Af ' 



and therefore, neglecting ;;*, 

 ^Adzl (s+j)2-r 



+ 



i' «^ _ (^—Stojr 



(2+"^r (2+^f (z+^f 



(IM — 112m;, +48«;,y 7(^4 — 8«;,)V. 



4(2 + Af 



4, /(^F\2 



1.2 



\dn) ~ {zJfAf (2+ J)* ^ '--^ ^« ^ 



■1 12w, 4 



4(2 + A) 



(IM— 112w, +48wjii^ 7(/J — 8«;, )V 



1 "T 



4(2 + ^)« ' 



_ , (^— 8«>,V 



(2+^)0 + (2+ Af 



£--8ttv)V. 

 4(2 + Af ' 



(F") 



(G") 



