164 BELL SYSTEM TECHNICAL JOURNAL 



dynamics of a particle. All constants, variables, and functions considered 

 here are understood to be real. 



Let Fit, X, y, z, p, q, r) be a function of the seven arguments indicated, 

 which, together with all of its partial derivatives of the first three orders, is 

 continuous in a region R defined as follows: 



ai < / < 02, 



bi < X < 62, 



R: Ci < y < C2, 



dx < z < ^2, 



p, q, and r unrestricted, 



the a's, b's, c's, and d's, being constants. 



Let x{t), y{t), z(t), (p{t), \p{t), and co(/) be continuous functions with con- 

 tinuous first derivatives, and let e, 77, and d be parameters, independent of /, 

 such that we have the relations 



b, < x{t) + eifit) < b,, 

 ci < y{i) + #(/) < C2, (oi < / < 02). 

 di < z{t) + 0co(/) < d., 

 Let Ti and To be constants, and let h and /o be parameters, such that 

 fli < Ti ^- h < T2 + h < 02. 

 We now consider the integral 

 /(e, V, 0. h, (2) 



= [ ' F{t, X i- e^,y-^ v^, z + Oco, x' 4- e<p', y' + rixP\ z' + do:') dt. 



It can be shown without difficulty that the integral exists and is a differen- 

 tiable function of e, ??, 6, h, ti. We are interested in formulae giving the 

 values of dl/de, dl/dr], dl/dd, dl/dti, dl/Sti at the point e = -q = 6 ^ t^ = 

 t. = 0. 



^ Since this section is purely mathematical, the constants, variables, and functions do 

 not necessarily have any special physical significance. 



"We treat the case of a function of seven arguments in order to fix the ideas, and 

 because this is a case we shall meet in Section VII. However, the discussion applies 

 essentially to other cases as well. In particular, in Section VII we shall also deal with a 

 case in which F has only five arguments, s and r being absent. 



