SECTIONAL TRANSACTIONS.— A*. 335 



If 1(0 > o, l(t') < o, and 



0(r, t') = S e»*S&i*/W) 



f(t,t')= Ji e-«(M</Vi), 



(x quadratfrei 



then the number of representations of the integer v as a sum of s integer 

 squares and r total positive quadratfrei integers of the corpus 



As,, (v) = 4j e-"' 5 w<*) yy e* (* + 2 «, f + 2M ')y> (« + zu, t' + zu') . 



_ e ~2rriS{ V ulV<l) d u du' , 



where E is the parallelogram in the (u, u')-plane defined by 



u = xo^ + yco 2 , w' = axV + yo 2 ', — £ < * <£, — £ < y < £■■ 



(coi, co 2 a base of & (\/d) ; to/, « 2 ' their conjugates) and we take t = 

 i/VNv, t ' = - i/i/Nv. 



We divide up the domain of integration E into regions F y by means of a 

 Siegel F-dissection of order (A r v) c , where o < c < 1 and the actual choice 

 of c depends on r and s. Then 



A,-,, (v) = * e -«S(w/vd) 2'// Qsfr du du ' } 



where runs over incongruent y (mod. E) whose ideal denominators a 



are such that Na < (Nv) c . The crucial step is to replace 0, / by suitable 

 approximation functions in the domains F y . For we use Siegel's 

 estimation 



e -KJNaVNv\v\\v'\ 



y/(Na\v\\v'\) ' 



6(« + zu, t' + zu') - Jty— < K } 



for /we use the estimation 



f{t + zu, f + zu') - ^ -V S **> 



D 



6 2 = o mod a 



< K 3 Na (\v I + |o'|) 2Vv* + « + X- 4 Na iVv* + « 



here v = f + 2(m — y), o' = *' + 2(2*' — y')i so that a, 7/ are small near 



the point (y, y'). The final result is : 



_ s — 3 



For o < r < — — 

 = — 2 



A -' (v) = i>+T*)* + *-* Nv ' + * s ~ ' s ^ + ° (iVvr,+ }s ~ }) 



as iVv -> 00 , where 5(v) is a convergent ' singular series ' whose sum depends 

 on the arithmetical properties of v but lies between two fixed positive 

 constants. 



Monday, September 14. 



Discussion on The theory of complex atoms (10.0). 



Prof. D. R. Hartree. — Non-relativistic treatment of electronic struc- 

 tures of atoms. 



' Solar-system ' approximation, in which mutual forces between electrons 

 are first omitted and then included, is a perturbation, too crude to give 

 significant results. 



