no 



REPORTS ON THE STATE OP SCIENCE. — 1916. 



§ 5. The corresponding combinations of the ker and \iei functions are : 

 Xk{x) =ker2a; + kei2a; =Ko{x vDGoix^l) 



V/fc(a;)=ker'2a; + koi'2^ =K'oG'o 



Zk{x) =kev X ker'x- -H kei x kei'a;=-(K'uGo + KoGr'o) 

 WA;(a;)=ker x kei'ic — kei a; ker'a; = =,-(K'oGo— KqG'q). 



§ 6. Mixed (ber, ker, &c.) product-functions arise^from 



lo{xs/l)Go{xs/~i) &c. 



and the real and unreal parts of this product will be called Xr(x) an'd 

 Xu{x) ; and there are corresponding combinations analogous to the V, 

 W, and Z functions already given : 



Xr(a;) =ber a; keric+beia; keia;1 y^/^\_l y^,/'^\ t /^ /np f„ r\ 

 X«(4=ker X bei a; - ber a; kei xj Xr(a;) + .Xt.(a;)=Io(a;s/OGo(a; ./.) 



Vr(a:) =ber;a; ker'a^ + bei'a; kei'a;-| Vr(a;) + tV«(^)=I'oG'o 

 Vtt(a;)=ker'a; beia; — ber a; kei .■» J ^ ' ^ ' " " 



Zr{x) = (ber'a; ker a; + ber a; ker'a; + bai'a; keia;-|-bei x kei'a; - 



Zzt(a;) = 3(kera; bei'a; + ker'a; bei.r — ber'a; kei a; — ber a; kei'a: . 



Zr{x) + tZu{x) = ^^{l'oG, + loG'o). 



Wr(a;) =- (ber x kei'a; + ker x bei'a;— ber'a; kei x - ker'a; bei x - 

 A 



Wtt(a;)=-(ker'a; ber a; + bei'a; kei a;— ber'a; ker a; — bei'a; kei a; - 

 A 



y^r{x) + t-^u{x)=l (IoG'o-r«G„). 



At 



§ 7. The last four may be simplified by the following relations, which 

 arise from the well-known property of the Bessel functions : — 



Io(a;)K'o(a,-)-ro(a;)Ko(^)= --• 



Putting x\/iiot X and equating real and imaginary parts, 



ber a; ker'a; -f- bei'a; kei a; — ber'a; ker a;— bei x kei'a; = 



1 



ber a; kei'a; -f bei a; ker'a;— ber'a; kei a;— bei'a; kera;=0. 

 § 8. Hence 



Z7-(a;) =bera; ker'a; -f- bei'a; keia; + ^- =ber'ker a; -J- bei x kei'a; — ;r- 



2a; 2x 



