MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
027 
( 2ft + 1) (P , i + 1 Q„ — P /t Q„ + 1 ) = (2ft — 1) (PaQw-i — P»/_iQ«) 
—PiQo p oQi 
= 4{EF-F(F-E')} 
= 4(EF+FE'-FF) 
= 2n 
Again 
2S (P'„Q«—P„Q'„) = (2ft +1) { Q„(P« +1 —CP„) — P„(Q«+i—CQ„)} 
—(2ft+ i)(P m+ 1 Q„.—P«Q//+i) 
— 77 
In a similar way (y) may also be proved. 
10. As bearing on the question of the convergency or divergency of series occurring 
in any investigation it will be important to consider the values of P„.Q„ when n is 
infinite. Taking the expression for P„ 
it is clear at once that 
P.= 
"(C-S cos 6)^d6 
0 
P«+i< (C+S)P s >P* 
Further since P ;i increases with u, ^ is positive, hence P„ +1 >CP /; 
Also from 
cW 
Q*= 
i 
2 71 +1 
(C + S cosh 8 ) ~ "2“ 
Q»+i, S Q« 
Also since Q« decreases with u, is negative, and therefore Q,,^ > CQ„ 
Cl it 
Hence 
P„ +1 Q, +1 <P„Q„ 
but tends to the limit unity, so that the series 
But the series 
except when 
2P«Q„ is divergent. 
2P rt Q„ cosft(yfi-a) is convergent, 
4 m 2 
