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



leading to the transformation 



The parameters must comply with the condition for zero angular range, namely 



2w r = p + g, 



and with the condition for the vanishing of the source-term in w. Subject to these, 

 the constants are arbitrary and are in theory adjustable to meet a corresponding 

 number of possible requirements. 



The range of adjustable parameters in *@. M seems, however, to be unexpectedly and 

 embarrassingly great, and there suggests itself a doubt whether the degree of 

 generality of the formula can be really so high as it appears to be. It is noticeable 

 that the m s parameters do not enter into the condition for zero angular range, and it 

 seems worth while to study the ^ 33 type of curve-factor more closely. 



When w is real and c > w > c, 



which is not formally a product or quotient of rational integral functions of w. So 

 far as this is true, ^,3 could not be expressed as the product or quotient of curve- 

 factors the squares of whose moduli are rational integral functions of w. But it must 

 not be forgotten that there is a curve-factor, namely ^ = w + (i(? c 2 )''', whose 

 modulus is a constant. The use of this gives a fractionalisation of ^ 23 , for it is readily 

 verified that 



2 ~ l 2 c 2 ) 1 '' 



wherein, as k 2 > c\ (c 2 k l ) 2 < c 2 , so that ^ 23 is a fraction whose numerator is of the 

 type of ^ 6 and whose denominator is ^,. Moreover, it is to be noticed that < ^ l is only 

 a particular case of ^ 6 . 



There is therefore no loss of generality in discarding ^ 23 from the category of 

 fundamental types, and in forming new curve-factors by combining arbitrary powers 

 of different forms of ^ only. 



Thus the curve-factor 



^ = n{w-k r +(iv 2 -c 2 ) 1 /'}"' (88) 



f 



is no less general than ^ 26 . It may be substituted for ^ in the transformation (87) 

 for the problem of the doubly pointed ship, with the same condition for zero angular 

 range. 



33. The formula just obtained obviously suggests the further step of establishing a 

 functional relation between n r and k r , and letting k, range over all real values subject 

 to the limitation c > k r > c. 



