f558 ) ■ p 



/j (ê) and ƒ2 (t/) both became 1 and so now we can take in accord- 

 ance with it : 



^; ('2) = f and ^x(-f) = |, 



so that the first integrals become: 



dv du H du bv H 



— . — = — — {v—uy and ^ • T^ = 7^ {v—u)\ 



or 



ÖM bv Q bu bv Q 



bï] bt] 2^ bè b^ 2 ' ' 



By replacing moreover v — u by s^ and v -{- ic by s^ the final 

 equations become: 



©=(IJ-=«--KSJ=(&J+^«-- ■ " 



These are still to be solved. 



Mathematics. — "On the multiplication of trigonometrical series." 

 By Prof. W. Kapteyn, 



1. If ƒ (a) and if ix) are two functions which are finite and conti- 

 nuous in the interval from x = to x = jr, we have 



f {x) r= \ a^ -\- a^ cos x -j- «a cos 2, x -{-... . 

 f {x) ■=. h^ sin X -\- h^ sin 2 x -\- . . . . 

 (p{x) z= i a'o -|- a\ cos x -)- a\ cos 2 x ... . 

 (fi{x) = b\ sin X -\- h\ sin 2 x -\- . . . . 

 where 



2 /-TT 2 f . 



a,i =: — I ƒ (to) cos WO) dci) hn =z — I ƒ (o)) sin mo dm 







2 r^ , 2 r^ . 



a'„ = — I ^(to)co5nco(/(t> &'n = — I (p {(o) sin net) do). 







In the same way the product ƒ (.-?;). ^ (.2?) may be developed, this 

 product being finite and continuous in the same interval ; therefore 

 f{x) . (fi {x) ^ \ A^ -\- A^ cos X -\- A^ cos 2 X -\- . . . . 

 f{x) . (f {x) ■= B^ sin X ■\- B^sin2 X -\- . . . . 

 where 



2 r« 2 r^ . ~ 



An-=- — I ƒ (w) y (<^) coa nco cZco, ^n = — I /(w) ^ (to) «in ww cfoi. 







