590 Pbofesbor CHALLIS, ON A MATHEMATICAL THEORY 



only way in which a particular value of / is deducible from the general integral (17) without 

 assigning arbitrary forms to the functions F and G, is to suppose F{u) and G(v) to be 

 arbitrary constants. Let, therefore, F (li) = c, and G {v) = c. Then 

 F, («) = cu, G, {v) = c'v, 



cu- „ , . c'v- 



F, (w) = — , G, («) = — , 



Sic. = &c. &c. = &c. 



e- e' 



Hence /= (c + c') (1 - euv + -r^, uV - ^, _ ^, ^ g, • u'v' + &e.), 



= (c + c) (1 - 6,-= + jf^, - Y^^^. + &«:•) ('9)' 



by putting r' for .i- + j/'. Determining the arbitrary quantities so that / = 1 when r = 0, we 



have c + c' = 1. Also -^ = when r = 0, and —3 = - 2e. Hence / has a maximum value at 



dr **'"" 



the axis of sr, and is a function of the distance from that axis. 



10. It appears, therefore, that the required form of / is derived from equation (12), by 

 supposing / to be a function of .r^ + y''. That equation accordingly becomes, 



-4 +-4 +4e/=0 (20). 



dr' rdr 



Equation (19) is the integral of this equation in a series, the only mode in which it appears 

 to be expressible. By putting /= 0, we have for determining the corresponding values of r the 

 equation, 



= \ - er' + -,- , + &c., 



1- .2^ l^ 2-. 3" 



from which it appears that there are an unlimited number of possible values of r for which 

 / vanishes. Since there is no lateral propagation, the motion does not extend beyond a certain 



limitinff distance from the axis, at which f and -— both vanish. It is not, liowever, apparent 

 " dr 



from equation (20) that these quantities may vanish together, that being an approximate equation 



which does not give the exact value of -^— when /= 0. To ascertain whether tliis will be the 



dr 



case, recourse must be bad to equation (12) in the Paper on Luminous Rays, (p. 368), which 



was obtained without neglecting small terms. On putting ie for /cm' that equation becomes, 



f f 1 

 TF^lF-''7 = " ^''^- 



Assuming now that / is a function of r, we obtain, 



d'f df f df ^ 



•' dr' dr- r dr 



whence it is clear that if f= 0, ~ also vanishes. Since ~- = both when /= 1 and / = n, for 

 •' dr dr 



