Theory of Faults in Cables, 175 



(53) and (52), the bracketed quantities are always divisible 

 by a^ — al, and u is expressible as 



K=r 1 ^ 1 (a l3 &i)0i(a 2 > h) + r 2 ^ 2 («i, &t)^g(«s, &s) + . . . ; (56) 

 ?'. 6. in the form of the sum of a number of products, each 

 being a function of % and &, multiplied by the same function 

 of a 2 and b 2 and by a constant r. 



Then assuming 



Ex = 2A<fr(a, 6), E 3 = 2A<fe(a, 5) . . . , 

 it follows that 



j (/>) sin (^ +5) «to-nErfi(«, JJ-fAWo, 6)-. . . 



A=^° 1 li ^ • (57) 



if sin 2 (y +6) «te-fi{^(«, J)} 2 - *- 2 {<M>, ?')P -• • • 



When there are intermediate conditions, producing discon- 

 tinuity in v or -=- &c. at certain points ss ly a: 2 , &c, each sec- 

 tion must have its own series of the form (51). The a's are 

 the same for every section, being determined by the resultant 

 of all the conditions. The A's and 6's are different for each 

 section. Thus 



f(x) — 2 A sin ( ~y + b) from x = to x = a\, 

 = 2 A' sin ( -j + V \ ssx os 2 , 



The intermediate conditions enable A', b', A", b" , ... to be 

 expressed in terms of A, a, and b. If 



, 1 C* Xl • f Cl \ X 7 \ • /#2# , , \ , 



u'= j- I sm / -=— + Oi I sin I .-4- + 6> 2 ) «# 



+ f X2 ^ sin (?f + &',) sin (M + ^ + . . - . j 



then m' may, as before in the case of u, be put in the form (56), 

 and the value of A follows. 



+ ...-2Er0(a,6) 



A _ fffi<H7 +b )* + W*A*MT + >'>* m 



+ ...-Sr{#«,6)}i 



