ELECTRICAL PROBING WITH AN APPARATUS IN WATER 



189 



In a more general form ^ve can write 



■| {pn-l + 2) 



Qm^^JSoI 2j ^M ^ + 2 2j ^n COS ncpj' ] + 



V 



^{pn-1+2] 





(32) 



since the terms 6^ 1 + 2 2 ^n ) ^^^ ^2 1 "i" 2 2 ( ~ l)"^/2 1 ^^e partial forms 

 for kj' = + 1 and kj' === — 1. The same can be said for the terms 



W 



^Po+ 2(^n-p„+^n+p„) 



id 6o 



^P„+S(-im-p.+ ^n+p,) 



We will consider in a general form one of the terms in the second sum of 

 the expression (32): 



^Po+ 2 ^^n-p,+^n+pj cos n9? 



/p,+ (L(p„_i)+/(i+p,))cos9P + 



+ (/_{p„_2)+ ^2+p.) cos29?+ ...+ (Z_j+ Z2p^_i) cos(/Jo-l) 9? + 



+ (^0 + kp) cos Pq(p + (Zi + /gp^+i) cos {pQ+l)(p-\- ... 



Since the function /,- is an even function ^vith respect to i, then assuming 

 that L — I ;, we find 



rPo + S ^n-po + ^n+p„) COS nxp = Zo COS po(p + Zi [cos (/>o + 1) 9? + 

 L n = l 



+ cos(/>o — 1)9?] + Z2[co3 {pQ+2)(p+ cos{pQ — 2)g)]+ ...+ 



+ ho [^°^ 2^0 9? + 1] + ^p„+i [cos (2/?o + 1) 9? + cos 9?] + . . . = 



= cos/>o9)+ 2ZiCos/)q9?cos 9?+ 2Z2C03/>o9'co3 29?+ 3Z3Cosj9o9' cos 39? + 



+ ... + 21 p^cos Pq(p cos Pq(p + 2lp^_^^cos PqCP cos {pq-\- 1) 9?+ ... = 



= cos PqCP [1 + 2(Zj^cos (p+ Z2Cos29?+ Z3Cos39?+ ...)]. 



Thus we find that 



r 



^Po+2(^n-po+^n+p„)cosre«; 



COS Pq(p 



1 + 2 2 ^n cos n(p 



■ (33) 



