480 PROCEEDINGS OF THE AMERICAN ACADEMY. 



extended momentum of all the energy emitted by the electron, between 

 the ends of the segment ds of its locus. We wish to evaluate the 

 integral 



f%IS=ds [''a'k '^J^^^ MS. (117) 



J dS J R^ Vl — 1,2 



This integration may be simplified by the observation that the vector 

 dg is not only independent of the direction of the planoid which cuts 

 the boundary of the elementary tube in the surface dS, as has already 

 been shown in general, but is also in this case independent of the 

 position of the planoid, for dg/dS varies as l/W and dS varies as R^. 

 The integral therefore is the same for any planoid whatsoever, and we 

 may therefore choose for simplicity a planoid perpendicular to the 

 locus (/s, and cutting the hypercone in a spherical surface of unit 

 radius, that \s R = U = I. Substituting the value of a from (111) 

 gives, since i; = and 1 = i?(w— n), 



J ~|f/S = dsfe'-inxcT~(w — n)dw, 



where c/co is a solid angle at the center of the sphere subtended by dS. 

 The vector c, normal to w, is then along some diameter of the sphere; 

 and n is directed from the various points of the surface toward the 

 center. For diametrically opposite points the terms (cxn)^ n cancel. 

 We need only integrate the terms (cxn)- w. If the diameter deter- 

 mined by c be taken as polar axis, these terms may be expressed as 

 c^sin^^W; and the element of surface is sin 6 dddcf). The integral is 

 therefore 



dg ^ ^ e-c-'wds. (118) 



/ 



This integral should be the vector of extended momentum for all 

 the energy emitted by the electron between the two points considered, 

 and its projections on any chosen time and space should be the corre- 

 sponding energy and momentum. If the k4 axis is chosen parallel to 

 ds, that is if the electron is considered momentarily at rest, we obtain 

 a simple expression; for then w = k4, c- = V'V, and ds = dt. The 

 momentum altogether is zero, and the energy is 



^e^{V'V)dt. (119) 



o 



When some other k4 axis is chosen, such that the electron is assumed 



