464 DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



k respectively. The signs of these four values of F (w) are all positive if k > c, 

 and all negative if k < c. But if c > k > the signs are +, + , , +, and if 

 . > k > c the signs are , +, , . 



Thus, if k 2 < c 2 , F (w) changes sign either between w = c and w = + oo or between 

 w = c and w = ; accordingly ~F (w) has a zero and is not a curve-factor. But, 

 if F > <- 3 , Y(w) does not change sign on either of the above-named ranges of w, and 

 so lias no zeros. Thus for 1 F > c 2 there is a curve-factor 



V M = *-w+(>-e)* ......... (77) 



having the linear range +c to c, the angular range zero, and the order at infinity 

 zero. 



31. On examination it will be found that some of the curve-factors already 

 discussed are resolvable into factors of simpler type. If in ^ the parameter k be 

 replaced by the introduction of ft, such that 



k sinh a = c sinh fi, (a > ft > a), 

 it is readily verified that 



^ ; , = ir sinh a c. sinh ,8+ (ir c 2 )' 1 ' 1 cosh a 



= (2c)- l e*{w + ce-^ + (w 3 -c*) l/ }{ce"-*-w+(w>-c !i y i >}, (78) 



so that ^ 5 is equivalent to the product of two factors, one of the type of ^ 6) the 

 other of the type of ^.^. 



Similarly if in ^ 7 a parameter /3 be introduced such that 



k cosh a. = c cosh /3, (a > /3 > 0), 

 it appears that 



^ 7 = w cosh a- () c cosh p+ (tf-c*)'!'- sinh a 



- ^} t . (79) 



so that ^ 7 is equivalent to the product of two factors, one of the type of 9 t , the 

 other of the type of r $ 23 . 



V n and #, reclassified according as X is greater or less than unity, are equivalent 

 to 



<g> 24 _ w cosh a + c cosh /8+ (tg'-c 8 )'^ sinh a 



w + ccosh(/3-a) ' ..... ' ' 



in which > a, and 



^ 26 _ w sinh q-c sinh 0+ (w*-<?} l! * cosh at 



w-ccosh(/3-a) ...... 



