251 



^ \ cosh 2p (Jj - A co, 2p (^ - i\}\co.h 2p (j-i) <=os 2p (j - i V 



// coshp cos p \l cash p cos p 



lil (lie SHiïie way 



Urn K/ -C-f] ! K{a,ï,a) = 



r-^mi 



1 



<ih2q 



X 



fiinhóQ I ^ I Sin 



A sin 2 



^(t-OIH'K^-'O ^"'"'^(t-OI 



i f sin/l q sin q ;( sink q sin q 



Putting 



Vo (-«^t - 77-.. Y.(.'') = -7 I ^ + ^ i , 



(// l/i I COttipn COS pn 



21/3 



^'0 (^ ) = 7777 (^ — *). 4'« {■') = 77-, I r^^ + 



I sinA 2q„ ( 7 — 2 j «*'" ^?« ( 7 ~ M I 

 l\/l ' l/M sinh q„ sin q„ ' 



(« = 1,2,...) 



the functions 'f „ (.r), t|>„ (j;) (?i ^ O, 1 , 2 . . . .) will be the orthogonal 

 and normal cliaiacteristic numbers; they satisfy equation (2), ^ being 

 replaced by ihe conesponding cliaiacteristic number. 



Now drawing graphs of the functions y = tgx, y ^=. tqhx and 

 y^ — tghx in one figure, it is easily seen that /j„ is an angle in 

 the 2/i-''' quadrant, and ry,, an angle in the (2?j -|- 1 )-»'' quadrant. 

 For ?/ — »• 3D />„ and 5',, converge to Ihe iniddlepoints of the intervals. 

 From this it follows that cos p,, and sin p,, converge to ±il 2 and 

 it is easily seen that the absolute value of cp,^ [x) and tp„ [x) remains 

 less than a number which is independent from x and n. Now as 



Lim — = Lim — =:= jr 



n — >(» n ?i— >Qo n 



the two series occurring in 



A (x, i, >.) = h -^ + 



+ 1^'"<-^)^'"^-). ... (3) 



will be uniformly convergent and the right hand side therefore will 



be equal to K{x, B, )■). 



17 

 Proceedings Royal Acad. Amsterdam. Vol. XXVI 



