ON THE CALCULATION OF MATHEMATICAL TABLES. Ill 



Zu{x)=ker X bei'o;— bera; kei'a;=beia; ker'aj — kei x ber'a; 

 Wr(a;)=beraj kei'x— ber'o; keix=kerx bei'a; — ker'o; bei x 



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



2x 2x 



It will be noticed that jQ{x\/l)Ko{x\/~i)=X'r{x) — iX.u{x), &c. 



At the present time Yr{x) and Yu{x), called by Dr. Russell* 8{x) 

 and T(x) are the only mixed functions which have arisen in practical 

 work, and tables of these two only are included here, 



§ 9. As the four X- functions arise from the products IqJo, KqGo, and 

 IqGo (argument x-^ i), they must each be related in the same way to 

 their derivates ; and we shall now show that they are four independent 

 solutions of a linear differential equation of the fourth order ; as are also 

 the four V, the four Z, and the four W functions. 



§ 10. Differentiation of the X-, V-, Z-, W- functions. — The argument 

 X will be understood : X stands for Xb{x) or any of the other three X 

 functions, and V, Z, W, for the corresponding V-, Z-, and W- functions. 



Noticing that ber"a;= — - ber'x— bei a; and bei"a;=— bei'ic + berxt 



X X 



X'=2Z V'=2W--V 



X 



Z'=V-^Z w= x-^w. 



X X 



Further differentiation gives : 



Y" + h'=2X = 2(w-h^Yj] 

 X \ X J 



multiplying by x and successively differentiating 



V"+i+^+^V"=2X"-'+"~-^X"-2=2fw"+-W"-i^ 

 X x \ X J' 



Again X" + -X' =2V. Multiplying by x and successively differentiating 



X"+^ + -X"=2('v"-'+"~'''V"-'V 

 X \ X }' 



By these relations the successive derivates of X, V, W, and Z (which 

 =^X') can be calculated. 



* Alternating Currents, loc. cit. 

 t Or we may obtain our results from X6(a;) = I„(a; -v^j) J„(a; ^^i), Ac. (| 4). 



