KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58. NIO 3. 13 



(5) j7 =L= Jd£<p (f^=j £ 3 = 3r/ ; (l - H) + rV = 3r, ; + r»'^ - 8,), 



(6) 



1 ( d|y (-|— ^-) £ 4 = 3(1 — r-f + 6(1 — r-)rhf + r*r; 4 = 3 + 6r»(if — 1) + r*(ij* — 6r y 8 + 3 



Vi — r* J W l — r 8 / 



But we know that when we put 



(7*) ^ -«,). »M. 



i?<(^) is a polynom of the fcth degree. It is called the polynom of Hermite of the 

 t:th order. We have, indeed, 



Ä, (»;) = -',. 

 Ä,fo) = if-i, 



Ä, (»;) = — *, s L 3/,, 

 Ä 4 (^) = ^-6f + 3, 



(7) Ä 5 (i?) = — »/' + 10 1 ; 3 — 15»,, 



Ä 8 (»/)=J? G -15r/+ 45r/>-15, 



B 7 {r}) = — r + 21 ♦ ;' — 105 i; 3 + 105 », , 



B a [t]) = if — 28 </ + 210 i, 4 + 420 if + 105 . 



To obtain the successive #<(>,) the recursion formula 



may be used. It is at once found if (7*) is differentiated and we remember that 

 obviously R t (??) = —rj. 



Now, at present, we only need the first four of equations (7). Multiplying 

 (3), (4), (5), (6) by q> (rj) we arrive at the following important equations, 



(8) /fl-_ r ^(,)^(,)._ r *&k>, 



(9) Ifl = ftfa) (1 + r'Ä 2 (,;)) = cpj, t ) + f^Ä 



(10) ^-^(,)(- 8 r^(i ? )-H^(,))--Sr^^-H^L ) . 



(11) /« -9.(i?)(» + «HÄ,fo) + r*Ä«fo)) - SftO?) + 61*^0 + r*^3). 



