666 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(b) If Fis to be lamellar, we may write 



V=h v -r(v), (7) 



where t is any ordinary function. Its divergence is 



T (*) • v 2 (f) + Vt'(*0. 



If F is to be solenoidal as well as lamellar, we may obtain Lame's 

 condition immediately by substituting the value of Ffrom (7) in (5). 



(c) If, like the vector which defines the field of electromagnetic force 

 within an infinitely long cylinder of revolution which carries lengthwise 

 a uniformly distributed, steady current of electricity, V is solenoidal and 

 a function of u only, we must have 



-R^ia)]-' 



2-^(v)=^p (8) 



(d) If, like the attraction within a homogeneous, infinitely long, 



cylinder of revolution, V is lamellar and a function of v only, the gradient 



of v cannot involve u, so that 



9h 

 h v =f(v), where/ is arbitrary, or -~ = 0. (9) 



V 



(e) If F"is lamellar and a function of u only, y- must be independent 



fi v 



of u and 



9K 



C/U 



— is a function of u only. (10) 



n v 



In this case h v is either a function of u only or is expressible as the prod- 

 uct of a function of u and a function of v. 



(f) If Fis to be solenoidal* and a function of v only, the expression 



9K dV 



9v_ V 2 Q) dv (11) 



* See equation (27). 



