166 BELL SYSTEM TECHNICAL JOURNAL 



It follows, therefore, that we have 



An important special case is that in which ti and ^2 are zero (so that the 

 limits of integration are fixed), and 



<p{Ti) = cp{T,) = ^pm = yp{T,) = co(ri) = coCra) = 0. 

 In this case we have in general 



/(<, ,, e, 0. 0) - 7(0, 0, 0, 0, 0) = « Q ,(0 [^ - 1^ g A 



/"■',, fay "i a^1 J, , „ /"" /Afaf rf aj 



+ o{t, 7], e), 



where o(6, ??, 0) denotes a term, the exact form of which is unimportant, 

 which is such thai the expression 



o(e, 77, 6) 



approaches the limit zero as e, tj, and 6 tend simultaneously toward zero. | 



In particular, if the functions .r(/), y{t), z{t) satisfy the system of differen- 

 tial equations 



ddF_dF^ d dF _dF ^ ddF_d_F^^ 



dtdx' dx ' dtdy' dy ' dt dz' dz ' ^~ ^ 



we have (for all choices of the functions (p, \p, co subject to the conditions 

 stated) 



/(e, 7j, d, 0, 0) - /(O, 0, 0, 0, 0) = o{e, t?, 9). (21) 



